Two goals in two second-half minutes from on-loan striker Lauri Dalla Valle condemned Reds to successive League One defeats for the first time this season.
The Finnish striker who is on loan struck on 61 and 63 minutes and left Crawley to reflect on missed opportunities earlier in the contest.
Richie Barker named the same starting XI as that which had lost to Yeovil on Tuesday and they nearly got the dream start.
With just a minute on the clock Mike Jones back-heeled a dangerous Nicky Adams' cross that was cleared off the line and when the ball fell to Gary Alexander his drive on the angle hit the outside of the post.
It was a lively with Crewe threatening in the fifth minute when Mathias Pogba sliced his shot wide after Byron Moore had dispossessed Nathan Byrne on the right. Reds only cleared as far as Luke Murphy whose powerful drive was blocked with his legs by Paul Jones.
That was the only save Jones made in the first half as Reds began to get on top and forced Crewe to defend desperately on occasions. Billy Clarke had a shot blocked and Mark Connolly headed wide and then on the half-hour Alexander was left holding his head in his hands after another shot, this time from Hope Akpan's cross, was blocked by a home defender.
Akpan was next to be denied by some last-ditch defending and with Adams causing plenty of problems down the flank it was Reds who finished a goalless first half on the front foot although they lost Kyle McFadzean five minutes before half-time with Joe Walsh replacing him.
Crewe were always going to respond in the second half and so it proved and they could have been in front seconds after the break when Dalla Valle saw his header pushed away by Jones to Murphy who side-footed onto the bar when he should have hit the target.
That gave the hosts every incentive and Jones made another good save after 51 minutes this time to deny Pogba but ten minutes later the hosts were in front. Josh Simpson gave away a free kick on the left-hand side 25 yards out and when Murphy swung it in DALLA VALLE flicked it with his head from 12 yards and the ball sailed into the top corner with Jones helpless.
It got worse two minutes later. Aneke's pass inside Nathan Byrne was ideally weighted for Byron Moore to cross to the unmarked DALLA VALLE to fire home from eight yards to make it 2-0.
Reds switched to a 4-3-3 formation with Nicky Ajose coming on up front and he tested Martin with a low drive and the keeper made another save late on to deny Clarke.
At the other end Pogba headed wide from a decent position but those two quickfire goals had knocked the stuffing out of Reds and Crewe saw the game out relatively easily, leaving Town still in ninth place in the League One standings.