Two second-half goals in the space of three minutes from Rowan Vine, a third from substitute Dean Morgan and a late strike from Enoch Showunmi, after Richard Walker's consolation for Crewe, helped lift an evening hampered by gloomy conditions.
The game kicked off with thickening fog enveloping some attractive play on the field as Warren Feeney snapped Luton's first chance wide in the sixth minute from Vine's left-wing cross.
Crewe were happy to sit back, but showed they were still capable of an attacking threat when Gary Roberts fired just wide from 18 yards in the eighth minute after Kenny Lunt's lay-off.
It was Luton who dictated the early exchanges though and after Leon Barnett headed Steve Robinson's corner just over in the 15th minute, Showunmi showed a glimpse of his erratic brilliance four minutes later when he left two defenders for dead near the right corner flag before curling a left-footed effort just wide from 22 yards.
Steve Jones' 20-yard volley at Marlon Beresford saw Crewe register the game's first effort on target in the 23rd minute just seconds before Peter Holmes ended a jinking run with a left-footed strike that skimmed just past the post.
Alex defender Walker then saw himself almost score at both ends in the space of three minutes as he firstly headed over his own goalkeeper only for the ball to drift wide, before he prodded a Lunt free-kick past Beresford in the 27th minute only for Kevin Foley to clear off the line.
Despite Luton's possession, they didn't register their first shot on target until six minutes before the break when Vine's deflected 25-yard effort was comfortably held by Ben Williams.
Town pressed on after the break with the classy looking Carlos Edwards having a drive deflect over in the 48th minute before Holmes headed weakly at Williams after Showunmi's cross three minutes later.
Edwards and Holmes then saw efforts deflect wide before Steve Jones missed a glorious opportunity for the hosts as he blasted Michael Higdon's knock-down over from six yards in the 64th minute.
As frustrations began to grow, Showunmi set off on a marauding 50-yard run that ended with him sliding in Vine to sidefoot home from ten yards for his first of the season in the 69th minute.
Three minutes later Vine was upended in the box by Foster, who was sent off for his troubles and the ex-Pompey man stepped up to bag his and Luton's second from the spot.
Morgan beat Williams at the second attempt from Holmes' cross eight minutes from time before Walker headed home Lunt's free-kick two minutes later.
Showunmi wrapped things up when he registered from close range after Vine's pull back from the by-line.