The Alex struck the decisive blow in the 86th minute at Wast Hills, in a match which Blues could count themselves unlucky not to get something out of.
Blues made the early running in the contest, with Jack Storer firing just wide after five minutes before Greg Stewart saw his attempt blocked after good work in the visitors' box.
But with 25 on the clock, Crewe had a golden opportunity to take the lead after Luke Maxwell bundled over Callum Saunders in his own area and the referee pointed to the spot.
Daniel Udoh stepped up to take the kick, but his effort lacked pace or precision and was comfortably saved by Blues keeper Connal Trueman.
Having escaped with their goal intact, Blues began to ramp up the pressure.
Storer blasted the ball over from distance and then Steve Seddon went desperately close to breaking the deadlock when he met Josh Dacres-Cogley's low pass at the far post, but managed to strike the upright.
Blues were rightly disappointed when they had what appeared to be a perfectly good goal ruled out just minutes before the break.
Viv Solomon-Otabor (pictured) whipped in a neat cross for Storer to head home, but the referee adjudged he had impeded Alex keeper Dave Richards during the execution.
Richard Beale's charges continued to press after the break and Dacres-Cogley had his goalbound effort stopped on the line by Marcus Poscha after just 45 seconds.
Chances continued to fall for Blues, with Stewart, Noe Baba and Reece Brown trying their luck.
However, it was Crewe who were just inches away from opening their account on 77 when Callum Saunders' shot struck the bar, before bouncing down onto the Blues goal line and away from danger.
As the game entered its final ten minutes Blues captain Daniel Cleary looped a header just over the bar and then substitute Ronan Hale saw his volleyed effort blocked just yards out.
But it was Crewe that finally made the decisive breakthrough when a long ball was sent deep into the Blues area and Callum Aintey got to it ahead of Blues keeper Trueman and swept it home into an empty net.
: Trueman, Dacres-Cogley, Seddon, Baba, Cleary, Maxwell, Brown, Cooper (Hale, 65), Storer, Stewart, Solomon-Otabor.
Subs not used: Weaver, McDonald, Hale, O'Keeffe, Martin.