The win takes Pep Guardiola's team two points clear of Arsenal at the top of the table after 37 games.
Manchester City closed in on a fourth consecutive Premier League title by thrashing Fulham 4-0 on Saturday, while Burnley were relegated from the top flight after a 2-1 defeat at Tottenham.
Fifth-placed Tottenham's third successive defeat leaves them seven points behind fourth-placed Aston Villa.
Arsenal held on for a pulsating 3-2 win over arch rivals Tottenham in the north London derby to keep their grip on top spot in the Premier League.
Liverpool were held to a damaging 2-2 draw by Manchester United as they handed the initiative to Arsenal in the Premier League title race
Tottenham missed the chance to move into the Premier League's top four after a 1-1 draw at West Ham, while Newcastle were also held 1-1 by Everton on Tuesday.
Spurs have now come from behind to win in four of their last five home games and needed another second-half turnaround after Tahith Chong's strike put Luton into a shock lead after just three minutes.
Tottenham wasted a chance to move into the Premier League's top four as they slumped to a shock 3-0 defeat at Fulham
Injury-hit Liverpool moved four points clear at the top of the Premier League with a 1-0 win at Nottingham Forest, while Tottenham beat Crystal Palace.
Premier League strugglers Everton salvaged a 2-2 draw with Tottenham thanks to Jarrad Branthwaite's last-gasp equaliser, while Aston Villa moved into fourth place with a 5-0 demolition of Sheffield United on Saturday
Liverpool sit five points clear of second-placed City, who have a game in hand as they try to spoil Juregen Klopp's bid to bow out with the second Premier League title of his nine-year reign.
Manchester City had lost all five of their previous visits to Tottenham's plush arena without scoring a goal since the stadium opened in April 2019
Manchester United will be squaring off against Tottenham Hotspur for their next Premier League match on Sunday at the Old Trafford.
Eric Dier spent a decade at Tottenham and will be reunited with former Spurs teammate Harry Kane in the Bavarian capital.
Timo Werner returned to the Premier League on Tuesday after the German international joined Tottenham on loan for the rest of the season from RB Leipzig.
Holders Manchester City will have to end their hoodoo at the Tottenham Hotspur Stadium to progress in the FA Cup fourth round as non-league Eastleigh could face Manchester United.
Arsenal's Premier League title challenge suffered another blow in a 2-1 defeat at Fulham while Tottenham beat Bournemouth 3-1.
Arsenal missed out on the chance to return to the Premier League summit on Thursday, losing 2-0 to West Ham as Tottenham went down in a six-goal thriller at Brighton