The Imps were without inspirational skipper Callum Ward through suspension for the semi-final tie, which took place at the Pirelli Stadium on Tuesday evening.
The first quarter of an hour would prove to be vital for the Imps' defence against a very powerful Burton front line and they dealt with the opening period without giving the hosts any clear cut chances.
The first chance of the game came on 25 minutes when a good move found Rob Smith down the right channel who cut the ball back from the byline and it was inches away from Alex Simmons. Moments later an identical move saw the ball reach Cenk Acar who couldn't quite get his shot away.
Burton responded by winning a free-kick which saw full back Kieran Walker receive a booking, then against the run of play the deadlock was broken on 35 minutes when Burton's Joe Doyle broke down the right hand side and whipped a cross in which was finished superbly by Leandro Browne.
With the game finely balanced approaching half-time a defining moment came when a Burton corner was headed powerfully past Imps' stopper Henry Smith but Kieran Walker cleared the ball off the line.
City came out for the second period revitalised and an early spell of pressure paid off on 52 minutes when Matt Cotton pinched the ball high up the pitch, Matt Bowles broke into the box and squared for Simmons to smash home from close range.
City continued to press all around the pitch - Cotton cut inside from a corner and shot wide, Kegan Everington's effort from distance was saved, and Rob Smith's near post effort was smothered by the home side's goalkeeper.
Burton remained a threat from set pieces but the game appeared to be heading for extra-time when Simmons broke over the half way line and released Akar into the box and as Akar and the defender came together both players went to ground and the referee awarded a penalty. Once Burton's protests had calmed down Simmons stepped up to calmly slot home and put the Imps in front for the first time.
The drama hadn't ended as Henry Smith made an outstanding stop at his near post and the Imps defence, marshalled brilliantly by Nick Green, held out for a deserved victory on the night.
The Imps will now face Shrewsbury Town in the final after they defeated Grimsby Town 2-1 at Blundell Park on Thursday evening. Details of the final to be announced at a later date.
Imps: H Smith, Grundy, Walker, Everington, Cooke, Green, Simmons, Bowles, Akar, Cotton, R Smith. Subs: Helsdown, Harper, Love, Wright, Walton.