Substitute Luke Rodgers inspired the dramatic finish as he set up the goals for Kenny Lunt, who thudded in a superb diving header, and Luke Varney.
Mike Newell's side bossed the first half and took the lead when winger Dean Morgan brought down Ahmet Brkovic's cross at the far post and prodded a delicate finish across Ben Williams.
Crewe had already hit the bar twice via a Steve Foster header and a Mark Rivers shot, when teenager Billy Jones picked up a loose ball from a free-kick and rammed a 15-yard shot home.
Luton were always dangerous though and Crewe were grateful for a fine performance from goalkeeper Williams, who flung himself to keep out Peter Holmes' header and stretched to hold on to an effort from the dangerous Warren Feeney.
In the second half Brkovic sent an overhead kick into Williams' arms, but it was Crewe who looked more like bagging a winner.
Youngster Gary Roberts missed the far post with a thunderous 25-yarder, while Billy Jones also came close to a second when he let fly from the edge of the box.
Substitute Rodgers teed off the final flourish when he crossed for Lunt to finish with a spectacular header three minutes from time.
Then fellow replacement Varney ran on to fire past Marlon Beresford in the last minute to seal an important win for Crewe. It was their first since the 2-1 home triumph against Burnley on the opening day of the season.