An own goal from Luton defender Daiki Hashioka saw the home side take the lead in the second minute, before Mateo Kovacic and Erling Haaland netted to send the champions 3-0 ahead.
Luton had one back in the 81st minute Ross Barkley, but Pep Guardiola's side netted twice more in the final exchanges through Jeremy Doku and Josko Gvardiol to secure all the points in style.
The result has seen Man City move two points clear of second-placed Arsenal at the Premier League summit, while Luton remain 18th, one point behind 17th-placed Nottingham Forest in the battle to stay in the league.
Man City needed just two minutes to make the breakthrough at the Etihad Stadium, and it came courtesy of an unfortunate own goal from Hashioka, with a volley from Haaland, which was going nowhere near the goal, hitting the defender's face and flying into the back of the net.
As expected, all of the pressure was coming from Man City, with Luton goalkeeper Thomas Kaminski making a smart save to deny Gvardiol, who netted a spectacular goal against Real Madrid last time out, before Kevin De Bruyne fired over from the resulting corner.
Matheus Nunes then had a golden chance in the 14th minute after smart work from Doku, but the midfielder's effort was straight into the body of Kaminski.
Haaland was on the stretch in the 18th minute in an attempt to meet a cross from De Bruyne, and he could only poke wide of the target, as Luton continued to come under pressure, with Gvardiol also coming close shortly after with a header which was cleared by Tahith Chong.
Ruben Dias drilled one at the goalkeeper in the 26th minute, before Nunes fired against the outside of the post shortly after, and Luton were struggling to get out of their own half at the Etihad Stadium.
At 1-0, though, they were in the match, with the visitors waiting for that one chance to potentially level.
Julian Alvarez bent one over the crossbar late in the first period, before Kaminski was forced to make a save to deny De Bruyne, but Man City were unable to add to their early advantage, with the early own goal all that separated the two teams heading into the interval.
Nunes might have found Haaland for a tap-in early in the second period, but the midfielder slipped at the final moment, and Luton managed to clear their lines, before Alvarez fired over shortly after.
Dias and Alvarez then both struck just wide of the target in the period that followed, with Luton somehow remaining in the match despite rarely getting out of their own half at the Etihad Stadium.
Man City managed to double their advantage in style in the 64th minute, though, with Kovacic picking out the top corner of the net with a stunning half-volley from just inside the Luton penalty box.
Luton were almost back in the match soon after, but Cauley Woodrow struck against the crossbar after smart work from Fred Onyedinma, before Kaminski did brilliantly to deny Rico Lewis down the other end.
Kaminski did well to keep out Nunes in the period that followed, but not too much was coming back from Luton, with Man City in complete control of the contest against their relegation-threatened opponents.
The hosts were awarded a penalty 75th minute when Onyedinma fouled Doku, and Haaland stepped up to convert, as the reigning champions made it 3-0 on the afternoon.
Luton gave their fans something to cheer in the 81st minute, with Barkley squeezing one into the bottom corner after finding space inside the Man City box.
However, the hosts netted again in the 87th minute through Doku, with the forward driving into the Luton box before picking out the bottom corner with a careful finish, before Gvardiol got his name on the scoresheet in stoppage time to make it five for Guardiola's team.