Manchester City were 4-3 victors in Wednesday's second leg, but for the third straight season failed to reach the last four under Guardiola as two big VAR reviews went in Spurs' favour.
Mauricio Pochettino hailed his Tottenham Hotspur players as "heroes" on Wednesday night, after the London side dramatically secured a place in the Champions League semi-final.
Meanwhile, Manchester City manager Pep Guardiola insisted VAR does produce "fair football" despite a cruel quarter-final exit on away goals to Spurs, after a pulsating tie ended 4-4 on aggregate.
City were 4-3 victors in Wednesday's second leg, but for the third straight season failed to reach the last four under Guardiola as two big VAR reviews went in Spurs' favour.
Firstly, Fernando Llorente's decisive goal 17 minutes from time stood despite the ball appearing to gently clip the Spaniard's hand on its way into the net. The decision would've been harsh, however, given the touch looked accidental.
Then deep into stoppage time, Raheem Sterling had the strike that would have sealed his hat-trick and seen City through ruled out after Argentine striker Sergio Agüero was shown to be offside in the build-up.
"I support VAR. It's just to see the goal from Llorente is handball. From one angle looks like handball, from the angle the referee saw it doesn't," said Guardiola. "I'm fine for the fair football, for the fair decisions. If it's offside, it's offside. In the future, even the present, it will be fair."
Elimination ended City's quest for a quadruple and Guardiola lamented a series of costly errors across both games, after Agüero also missed a penalty in the first leg, that saw his side bow out to a side 16 points behind them in the Premier League.
"It's tough. We were close to go through, it's cruel, but we have to accept it," added Guardiola. "The mistakes in this competition punish you a lot. Unfortunately it was a bad end for us."
Unlike Guardiola, Pochettino has previously been a critic of VAR, but believes the decisions to award City a penalty in the first leg and not punish Fernandinho for a clash with Harry Kane balanced themselves out over the tie.
"You have to trust the VAR when it is against or for you," said the Argentine coach, whose star continues to rise. "The decision to include VAR in football is going to change the game."
Pochettino labelled his players as "heroes" for reaching the semi-finals of the European Cup for just the second time in their history, after a season in which they have not signed a single player and played the majority of the campaign at a temporary home in Wembley.
"On this type of night, this type of victory, only I can feel proud. I feel they are heroes. With all the circumstances to be in the semi-finals of the Champions League they deserve a lot of praise."
Guardiola's decision to not start Kevin de Bruyne in the first leg will be further questioned as the Belgian bagged a hat-trick of assists to take City to the brink of the last four. The first set up the first of four goals in 11 minutes as Sterling opened the scoring.
However, Spurs struck back twice in three minutes as Son Heung-min's double took his tally for the season to 20.Yet, with Kane expected to be sidelined for the rest of the season, Pochettino has another selection headache with Son suspended for the first leg of Spurs' semi-final against Ajax.
A kamikaze opening had another twist, granting City the rub of the green, as Bernardo Silva's effort deflected in off Danny Rose and wrong-footed Hugo Lloris at his near post to bring City level on the night.
By the midway point of the first-half, the hosts were back in front as Bernardo and De Bruyne played creators once more and Sterling finished off a low cross at the back post.
Spurs' hopes of holding on to their advantage on away goals suffered another blow when the influential Moussa Sissoko was forced off with an injury.
Pochettino chose attack as the best form of defence by replacing the midfielder with Llorente in what proved to be a decisive change.
De Bruyne had his third assist on the hour mark as he surged through midfield before timing his pass for Aguero to perfection and the Argentine rifled past Lloris.
Having gone ahead for the first time in the tie, Guardiola soon sent on Fernandinho to restore some sense of control, but City were pegged back again as an incredible tie swung back in Tottenham's favour.
Kieran Trippier's corner was bundled in by Llorente, but there was controversy as the ball seemed to hit the striker's arm on its way in.
After a VAR check, though, Turkish referee Cuneyt Cakir stood by his original decision and the goal stood.
And VAR went against City again deep into stoppage time as delirium soon turned to disbelief inside the Etihad when what would have been a hat-trick for Sterling was ruled out.
After a series of frustrating defeats when the stakes were at their highest, Mauricio Pochettino's Tottenham finally came through on the big occasion.
Falling short in FA Cup semi-finals, Champions League knock-out games and Premier League title battles had tarnished Pochettino's otherwise impeccable credentials as one of the brightest young managers in Europe.
But, playing without injured star striker Harry Kane, Pochettino inspired a tour de force performance from Tottenham.
Pochettino is yet to win his first major trophy, but he out-witted City boss Pep Guardiola, who has won every prize the game can offer.
Tottenham were aided by two huge VAR decisions that allowed to them to score the decisive third goal when Fernando Llorente bundled in off a combination of his arm and hip – and then escape what would have been a Raheem Sterling last-gasp winner as offside was given against Sergio Aguero.
But luck wasn't the reason for Pochettino's success as he leads Tottenham in the Champions League semi-finals for the first time despite not buying a single player in the last two transfer windows due to the restraints from the move to their new stadium.
Tottenham are in the last four of a European competition for the first time since the 1984 UEFA Cup because Pochettino and no wonder Tottenham's jubilant fans chanted Pochettino's name loudly as he danced with his players on the pitch at full-time.
Guardiola suffers more Euro misery
Pep Guardiola held his head in his hands when Ilkay Gundogan's close-range effort ballooned over the bar in the closing moments, the Manchester City manager's angst at his latest Champions League failure clear for all to see.
Guardiola is rightly regarded as the best manager of his generation after trophy-laden spells with Barcelona, Bayern Munich and now City.
But despite having access to vast financial backing at all three clubs and coaching some of the world's greatest players, Guardiola has only got his hands on the Champions League trophy twice in his 11-year career.
The Spaniard won it with Barcelona in 2009 and 2011, while also earning a winners' medal as a Barca player in 1992.
Dumped out by Liverpool last year, City, who have never won the Champions League, once again came up short on the big occasion in Europe.
City's bid for an unprecedented quadruple is over and now the question is whether they can recover in time to overhaul Liverpool in the Premier League title race.
Spurs could ruin that City dream as well when they return to the Etihad Stadium in the league on Saturday.
Son shines again
Son Heung-min has proved this season that playing without Harry Kane won't be fatal to Tottenham's ambitions and the South Korea forward was in sparkling form with a brilliant double.
With Tottenham trailing after just four minutes and Kane sidelined, Son set about dismantling City's defence.
Seizing on Aymeric Laporte's miscued block, Son quickly whipped a low strike that caught Ederson off balance as it bounced in off the City goalkeeper's foot for the seventh minute equaliser.
That was followed by Son's second three minutes later as he shifted away from Kyle Walker before curling a superb shot into the far corner.
Son, who had scored the first leg winner against City, has now bagged 11 goals in 11 matches with Kane absent this term.
His pace, skill and astute movement make him a different proposition for central defenders than Kane and some have even suggested Tottenham look better with Son up front than the England captain.
However, there was a sting in the tail as a second half booking for Son, who tugged on Kevin De Bruyne on the edge of the Tottenham area, means he will miss the first leg of the semi-final.