They trailed to goals by Dan Lokko (19 minutes) and Joe Robbins (38) before Nick Kinina netted on the stroke of the interval.
Blues, having had to readjust their starting line-up due to injuries in key areas, stepped it up after the break and forged good openings to at least get the equaliser.
But in the end Crewe stayed resolute enough to keep their lead at the Trillion Trophy Training Centre, Wast Hills.
An early chance came Blues way when Amari Miller reacted quickly to a header back to the 'keeper that lacked conviction but unfortunately his attempt to lift the bouncing ball into the net was thwarted by a combination of Jerry Le Grand Akpo and a defender.
Miller, coming in from the right, then struck a shot just wide of the far post.
Blues fell behind when a ball in behind bypassed Blues defence and Lokko slotted it low beyond Aaron Clayton.
Blues struggled to get a real foothold thereafter and Crewe doubled their advantage after passing through the thirds for Lokko to set up Robbins, who swept in an impressive finish across the 'keeper from just inside the area.
Blues lifeline came in the 45th minute after Tate Campbell was felled trying to force his way through towards the box.
Campbell himself took the free-kick and went low, past the wall to the keeper's side, and Akpo parried, only for Kinina to slide in first and put the loose ball in.
Blues, already without Yoane Zohore due to injury, had to withrdaw Oumar Traore at the interval as well.
Miller went up front in the attacking three and Josh Williams had to drop deeper into midfield as trialist Makai Murray came on, on the right.
Blues performed with more swiftness in their play and Kyle Hurst had a couple of good runs that almost yielded chances.
Remi Walker, who, like Campbell, dictated more and more of Blues better rhythm, curled a free-kick just over onto the top of the rigging.
And then Walker nicked possession in midfield and sent Williams in, down the middle. Williams struck a powerful shot, that Akpo grasped at the second attempt, and had it gone a few yards either side of the 'keeper it surely would have been a goal.
Crewe regrouped and came at Blues again. Clayton made a fine stop from a blasted attempt by Joe Levey and from the follow up attempt, struck as crisply, Kade Craig had got back in to help protect the goal and cleared from the line.
Blues remained urgent and after interplay between Murray and Campbell, Miller was teed-up and his right footer from 20 yards flew just past the post.
In the 74th minute a fine raking ball from left to right by Daniel Djurovic sought out Murray, who had got in round the back, but he couldn't get control or direction on his volley from a great position.
Moments later Tyreece Onyeka lifted a close range shot over the diving Clayton and onto the crossbar as the ball bounced around the Blues box from a Crewe raid.
Blues continued to strive to find that elusive second goal but the visitors managed to successfully shut down the avenues to the finish.
Blues: Clayton, Dugmore (Chang 84), Djurovic, Walker, Kade, Kinina, Miller, Campbell, Traore (Murray), Williams (Fortune 79), Hurst. Subs not used: Rouse, Nguepissi.