Dummigan fired into the side-netting before Gilchrist was first to the rebound after Gollini pushed the ball into the air to make it 2-2.
Once again, the Reds responded in the right manner with Daehli teeing up Love for a full-blooded drive that found its way past Jakovlevs to regain the lead. Any hopes that would be sufficient to clinch a place in the next round proved unfounded as Declan Dalley, who had replaced the injured Ioannou moments earlier, put Gollini under pressure with a back-pass and Gilchrist won the tackle with the keeper to slip his second of the night into an unguarded net.
Daehli tried to inspire a late winner but scooped over the top before flashing a dangerous ball across the face of goal. Wilson had the final attempt of normal time but his powerful angled drive was well held by Jakovlevs.
The Reds continued to dominate in the first period of extra-time but Pearson's shot was beaten away by Jakovlevs with the midfielder pulling up with cramp afterwards. Howieson gave Gollini an uncomfortable moment from long range but most of the chances were created at the other end with substitute Jack Rudge upset not to win a penalty and Wilson heading inches wide following a Januzaj centre.
Daehli was denied what looked a stone-wall spot kick after a mazy dribble but both teams looked leg-weary as the tie wore on. Out of nothing, Burnley conjured up a fourth goal with substitute Bradley Jackson getting away down the right to deliver a measured cross for Gilchrist to nod in for his hat-trick with eight minutes remaining.
Jakovlevs pulled off a remarkable double save to deny Wilson and Januzaj and then tipped a Liam Grimshaw shot around the post with his fingertips. Daehli bent a shot wide so there was no way back and last year's semi-finalists bow out at the first hurdle despite giving their all.
United: Gollini; Love, Grimshaw, Ioannou (Dalley 77), Rowley; Barmby (Rudge 55), Pearson (Byrne