The Cityzens were on a four-game winless run in the English top flight but sealed victory after goals from Bernardo Silva and Jack Grealish.
Pep Guardiola’s side entered the match without star players Erling Braut Haaland and Jeremy Doku, but they still made a good start to the Kenilworth Road encounter, with Phil Foden drawing a good save from Thomas Kaminski within the first two minutes.
Rodri then made a great effort from the edge of the box but the Spaniard was also denied by Kaminski, who sprang to his right to push it over the bar.
The Belgian goalkeeper again proved decisive in keeping the scoreline 0-0 when Foden unleashed a powerful shot from the edge of the box.
But then just before the break, Luton popped up with a stunning goal to send the home fans into ecstasy. Elijah Adebayo headed home the opener after a good delivery from Andros Townsend to the far post.
However, Man City came with all guns blazing in the second half and restored parity within a few minutes of the restart.
Bernardo Silva whipped the ball into the far corner after it fell kindly for him when Rodri was tackled by Tom Lockyer.
Jack Grealish added another just three minutes later, tucking past Kaminski after a brilliant ball from Alvarez.
Ross Barkley went on a jinking run and let fly a great shot but the effort went wide of the right post.
Luton manager Rob Edwards made two changes to make a comeback in the game, bringing on Carlton Morris for Townsend and Sambi Lokonga for Marvelous Nakamba.
But they failed to breach City’s defences and ultimately fell to a 2-1 defeat.
- Arsenal shortlist four strikers to replace injury-prone Gabriel Jesus
- Premier League reveal April fixture schedule for TV coverage
- Simone Inzaghi wanted by Liverpool, Man United & Chelsea – but who’s the best fit?
- Bowen hat-trick hands West Ham first win of 2024, eases pressure on Moyes
- Chelsea eye Amorim to replace Pochettino this summer
- Analysis: Liverpool did all they could to win the League Cup – and ultimately triumphed
- Everton ten-point deduction reduced to six
- Jurgen Klopp could win the quadruple this season and still not be Liverpool’s best-ever manager
- Chelsea & Man City target set to join Real Madrid in 2024 or 2025
- Cristiano Ronaldo makes obscene gesture at rival supporters chanting ‘Messi’