Fernando Llorente's strike proved decisive as Spurs edged a pulsating tie on away goals.
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Tottenham reached the last four of this competition for the first time since 1961/62 on a thrilling night at the City of Manchester Stadium.
An astonishing start spawned the quickest five goals in a match in UEFA Champions League history with the lead changing hands twice. Raheem Sterling produced two clinical finishes to put City 1-0 and 3-2 up, the latter after Heung-Min Son had struck twice in four minutes to give Tottenham a 2-1 advantage. Bernardo Silva's deflected effort within a minute restored parity.
Hugo Lloris denied Kevin De Bruyne and Ederson repelled substitute Fernando Llorente's header early in the second half, before City took the lead for the first time in the tie when Sergio Agüero rifled in their fourth. Spurs did not buckle, though, and Llorente diverted Kieran Trippier's corner past Ederson to settle a sensational game.
View from the stadium: Simon Hart
It was thrilling, it was breathless, it was a night nobody here will forget. It showcased City's attacking brilliance – and yet at the end of it, Tottenham are the team who can look forward to their first European Cup semi-final since 1962. The end-to-end drama began with a magnificent Sterling goal inside four minutes, the first of five goals in the first 21 minutes.
Son's double swung the tie one way but with De Bruyne pulling the strings you felt it was not over. It looked like being City's night when Agüero smashed in their fourth goal; yet Spurs delivered the sting in the tail and Guardiola's side's pursuit of this trophy must go on.
Man of the match: Heung-Min Son
Not just the key player in Manchester but across the tie, Son's display here – particularly in that madcap first half – set the tone for Tottenham. The forward's first goal may have relied on an element of fortune, but his second was a magnificent curling strike; Son also led the line relentlessly throughout and provided the crucial goals in the absence of Harry Kane. His fist-pump to the bench following his second here underlined just how much this meant to the Korean – and how much it means to the club he has fired to a maiden UEFA Champions League semi-final.
6: City have lost their last six UEFA ties in which they lost the away first leg.
7: Tottenham are the seventh English club to reach the #UCL semi-finals, with England overtaking Spain as the competition's most successful country in this respect.
18: Spurs have won 18 of the 20 UEFA ties in which they won the home first leg.
20: Tottenham have scored in their last 20 UEFA Champions League matches.
98: City have scored a total of 98 goals during their 26 competitive home games this season.
What does it mean?
Tottenham reach the semi-finals for the second time in their history – after a 57-year wait – while City are eliminated by a Premier League opponent for the second season in a row. Ajax await Mauricio Pochettino's men, the first leg taking place in London on either 30 April or 1 May with the return a week later.
City's next two fixtures could prove pivotal as they seek to retain the English title. They host Spurs again on Saturday before a visit to arch-rivals Manchester United on Wednesday evening.
Tottenham remain in a four-horse race to finish third or fourth in the league and secure a place back in this competition next term. Following the trip to City, three of their remaining four Premier League contests are at home.