Goals by Adam Stansfield and Liam Sercombe put the Grecians on their way before Tom Craddock pulled one back for the Hatters before half-time.
The visitors comfortably held on throughout the second period to stretch their unbeaten run to eight games and leave the hosts ever closer to a third successive relegation.
Exeter began in positive mood playing neat, controlled passing football and opened the scoring with the first attack of the game in the fifth minute.
Wing-back Sercombe found himself well advanced down the Grecians right flank and delivered an inch-perfect cross that saw Stansfield lose Keith Keane to head home through the arms of Dean Brill from six yards for his eighth of the season.
Luton didn't take long to respond as Sam Parkin's 50-50 tackle with Matt Taylor in the ninth minute saw the ball spill out to Chris Martin, who hammered a 30-yarder which flew off the top of the crossbar.
Ian Henderson then had an angled 14-yard drive parried by Paul Jones in the 12th minute before a free-kick from the same player five minutes later saw Craddock force another save from Jones with a far post header.
But Exeter doubled their advantage on the half-hour when home skipper Keith Keane's mistimed attempt to play offside allowed Alex Russell to burst into the penalty area and, after Brill had half-stopped the midfielder's ten-yard shot, Sercombe was on hand to fire into the roof of the empty net from three yards.
Luton pulled one back in the 39th minute when Gallen fed Henderson, who cut the ball back from the right by-line for Craddock to hit a low ten-yard shot under Jones for his seventh of the campaign.
Exeter should have scored straight from the re-start when Rob Edwards played Steve Basham clean through only for the striker to shoot tamely at Brill.
Instead the Grecians were left to breathe a sigh of relief two minutes later as Parkin thumped an eight-yard header against the join of post and crossbar after Ed Asafu-Adjaye's cross from the right.
The first opportunity of the second period didn't arrive until the 66th minute when Brill saved Basham's close-range effort after Stansfield's low cross from the right.
Luton struggled to create clear openings with Keane going closest in the 87th minute when his stretched volley bobbled wide from 12 yards after a one-two with Craddock.
Substitute Ryan Charles sent a ten-yard header wide in the final minute but it was too little too late for the hosts.