First half goals from Tinashe Chakwana, pictured above, and Dwight McNeil sealed all three points for Burnley as they made the most of their game in hand on leaders Blackpool to secure a 10th win in 12 matches.
Michael Duff's side started on the front foot and looked to have took the lead with just 10 minutes gone, when Chakwana teed up Tommy Wood in the area, only for the latter to be flagged offside.
However, minutes later the young Clarets did take the lead through a penalty. Following a foul on Chakwana, the striker stepped up and slammed in the first of the afternoon.
The visitors were dominating and after being awarded a free kick from a dangerous position 25 yards out, up stepped McNeil to curl in a delightful effort and double the lead.
Wrexham came close early on in the second half when Niall Freeman found George Harry just outside the box, who fired a shot just wide of Connor King's goal.
But it was the high-flying visitors who looked more likely to extend their lead, when Olatunde Bayode beat Doug Tharme before squaring for Chakwana in the area, only for keeper Owen Williams to brilliantly deny the in-form hitman.
Both Blackpool and Burnley now have three games remaining, with the Seasiders boasting a better goal difference in a race that seems certain to go to the wire.
Burnley: King, Egan, Wilson, Moss, Mpofu, McNeil, Shrimpton, Yari, Bayode, Wood (Cropper 85) Chakwana.