And then, deep into stoppage time, Raheem Sterling thought he had his hat-trick, but VAR again came into play, confirming that Agüero was offside in the build-up.
Thus it was Tottenham that went through to book a date with Ajax. And Coach Mauricio Pochettino roared in joy.
Manchester City had some consolation though. It was their first win against English opponents in Europe after five successive defeats.
That record will means nothing to Coach Pep Guardiola as his side still have just the one semi-final appearance, and that was before he arrived at the club, in 2016, despite all the hundreds of millions poured in by the Qatari owners of the club.
The lessons learnt
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 knockout 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