The home side, who are featuring in the end-of-season play-offs for the third year in succession, had to stage a recovery after Luton's David Oldfield had put the visiting side ahead after only three minutes.
No club in the country has featured as regularly in the play-offs in the 1990s as Crewe and Dario Gradi's team kept their nerve as they went in search of an equaliser.
Mark Rivers put the home team back in contention when he equalised after 53 minutes. He drilled home a low shot at the near post after Shaun Smith had provided the opening.
Crewe sensed they had a chance of victory seven minutes later when the visitors were reduced to 10 men. Julian James had already been booked when he was shown the yellow card for a second time and was sent off.
The home fans in the Gresty Road crowd of 5,467 were celebrating after 67 minutes when Crewe went into the lead, Colin Little scoring his first goal since the play-offs last season.
Luton had to fight a rearguard action for the remainder of the match as Crewe piled on the pressure and were lucky not to suffer a worse defeat when Dele Adebola fired against a post with nine minutes remaining.
Goals: Oldfield (3) 0-1; Rivers (53) 1-1; Little (68) 2-1.
Crewe Alexandra: Kearton; Unsworth, Smith, Westwood, Macauley, Charnock, Whalley, Little (Johnson, 82), Rivers (Tierney, 74), Murphy, Adebola. Substitute not used: Anthrobus.
Luton Town: Feuer; James, Thomas, Waddock (Marshall, 8), S Davis, Patterson, McLaren, Alexander, Oldfield, Thorpe, Fotiadis (Grant, 60). Substitute not used: K Davis (gk).
Referee: T Heilbron (Newton Aycliffe).
Bookings: Crewe: Murphy. Luton: James, Alexander. Sending-off: Luton: James.
Man of the match: Little. Attendance: 5,467.Reuse content