Enniskillen's Colm Bradley and Ronan McCabe helped themselves to a goal in each half to steer their side to the semi-final stages in a thrilling encounter at Brewster Park.
Their opponents, who have yet to play their county final, did well to recover from Bradley's first minute strike. After 10 minutes a close range goal from Stefan White, followed by four unanswered points gave them a healthy lead.
At half-time, however, Enniskillen were just a point behind and in the 42nd minute McCabe went to work.
The youngster's goal provided the platform for a most impressive display by defence midfield and his fellow forwards to which Burren had little by way of response. Having scored their last point by John Treanor in the 35th minute they had to wait until the 54th minute for their next and last score, a converted free by White.
Enniskillen: B Dooris; K Gunne, R Curran, M Lilly; N Cox, M Farry, R McCluskey; T Brewster (0- 3), P Brewster; O McShea, R McCabe (1-3), Simon Bradley; R Judge (0-2), P Fitzpatrick, C Bradley (1-1).
Burren: M Rocks; M Doyle, C Byrne, C Doyle; L Magee, B McKernan, G McFerran; S Ward, R Mussen; G Murdoch (0-1), J Treanor (0-3), A Woods, Jimmy McAlinden, A McGivern, S White (1-3). Subs: C Campbell for McFerran. P Bradley for Mussen. P J McAlinden for Jimmy McAlinden
Referee: M McGrath (Donegal).