The Railwaymen withstood a first-half onslaught from the hosts before netting with their only two shots on target.
Frenchman Julien Baudet dived to nod in after 53 minutes from a pinpoint free-kick to give the visitors the lead.
It was cruel on Carlisle who had seen their quick passing game only let down in the penalty box.
Paul Murray - back for Carlisle after a three-month injury - saw his goalbound effort blocked by his own team-mate Karl Hawley who was standing almost on the goalline.
Then on-loan teenager Joe Garner fired a shot just wide from a tight angle and moments later Hawley's snapshot on the turn looped just over.
Garner again went close from long range before Crewe's Nicky Maynard fired a 35-yard dipping volley against the crossbar.
The Cumbrians started well after the break and Hawley almost broke the deadlock with a stinging shot that Ben Williams did well to block at his near post.
Moments before Baudet's opening, Maynard went close again when he found space outside the box and fizzed a powerful shot just wide.
Baudet's goal forced Carlisle to press Crewe even further and they almost grabbed a deserved equaliser when Kevin Gray rose highest from a corner but his header dropped inches wide.
Carlisle probably knew it wasn't to be their day after Paul Thirlwell's agonising miss on 65 minutes.
Kevin Gall carried the ball 40 yards down the right wing, cut inside and laid off for Thirlwell but, with the goal gaping, his shot flicked for the defender and looped wide up the post.
Hawley flicked another chance wide with five minutes to play before Maynard crowned his excellent performance with a great solo effort.
The teenager picked up on Danny Livesey's mistake and jinked between two defenders.
But despite picking up a hamstring injury on the way, he hobbled on and smashed a low drive passed Keiren Westwood for his 16th goal of the season to send the visitors home with all three points.