A first ever professional goal from 18-year-old Ryan Charles on his home debut, a Matthew Spring penalty and a Lewis Emanuel thunderbolt gave the hosts an unassailable first-half lead.
Oldham, who had Reuben Hazell sent off for a professional foul in the 32nd minute, had gone into the game with loanee Richard O'Donnell in goal and only four players available to sit on the bench.
And the Hatters took full advantage of their opponents' situation to claim a first win since New Year's Day.
Luton started the brighter, dictating possession without creating a clear opening.
Veteran Paul Furlong did spring the Latics' offside trap in the sixth minute, but his pull-back from the left couldn't find the advancing Paul McVeigh.
Just as the Hatters' upbeat start appeared to be fading, the hosts took the lead out of the blue in the 20th minute.
Richard Jackson's cross from the right found Charles and the youngster showed delightful skill to juggle the ball past Stefan Stam and fire a low, left-footed volley inside the near post from eight yards out.
Oldham hit back with a 20-yard effort from Deane Smalley, which was pushed around the post by Dean Brill.
However, the visitors were left staring defeat in the face in the 33rd minute when Furlong was hacked down as he ran on to Keith Keane's long pass and Hazell was sent off.
Spring tucked home the resulting penalty for his tenth goal of the campaign.
Two minutes later it was three as Keane's cross-field pass released Emanuel who advanced in on goal and fired in a low left-footed drive from 15 yards that nestled in the net off the far post.
The visitors almost pulled one back in first-half stoppage time when Smalley ran in behind Sol Davis, but Brill superbly saved the one-on-one.
Luton almost got a fourth 50 seconds into the second period when substitute Sam Parkin laid the ball off for David Bell, whose ten-yard effort was parried by O'Donnell.
The Hatters were happy to play out the rest of the game, but almost let Oldham claim a consolation when Brill had to leap to tip Leon Constantine's 81st-minute volley around the post.