Dean Marney moved closer to a first 90-minute run-out, but couldn't prevent the Clarets slipping to a second successive defeat.
Marney made his third successive start in the Professional Development League as he continues his comeback from knee surgery.
And the midfielder enjoyed his longest outing yet, completing 78 minutes before being replaced by striker Tommy Wood as Michael Duff's side tried to find a way back into the game.
However, a goal in each half was the difference between the two sides on a chilly afternoon at the Alexandra Stadium as Crewe secured the win that took them above Burnley in the table thanks to goals from Alex Hurst and Lewis Reilly.
Burnley started the game brightly and came close when Jamie Thomas’ effort on goal was stopped by Alex keeper Ben Garratt.
However, the Railwaymen took the lead with 20 minutes gone when the ball fell to Hurst at the back post and the full-back slotted home from close range.
The Clarets went close to an equaliser 10 minutes into the second half when Tinashe Chakwana saw his close-range header go just over the top.
Thomas then had another chance to pull it level but Garratt denied the Welshman inside the box.
And the visitors' best opening yet came when Thomas’ cross was flicked on by Oliver Younger in the area only for Dwight McNeil to direct his header just over the bar.
And as Duff's side pushed for a leveller, they were sunk by a second goal eight minutes from time.
Reilly picked up the ball from outside the area and drilled it into the bottom corner to make the points safe and punish the Clarets' lack of finishing power.
“I think we had a lot of the ball, but we didn’t really hurt them in the final third and that’s something we have been working on in training as well," said Duff.
"I think Aiden has made one save and they haven’t had many chances but we have taken two really soft goals.
"But they were better than us in both boxes really and we didn't really get in behind and hurt them as much as we would of liked to of done.”
The Clarets are back in action on Thursday night when they take on Colchester United in the Premier League Cup at Curzon Ashton (kick-off 7pm).
Burnley: Stone, Birch, Flowers, Younger, Koiki, Bayode, Marney (Wood 78), Ulvestad (Nabi HT), Chakwana, Limb (McNeil 63), Thomas. Unused substitutes: King, Clarke.