Ben House's first goal of the season sent City into the third round of the Carabao Cup.
House's curled effort in the 89th minute, after coming off the bench, sealed victory at the Pirelli Stadium.
It was a deserved victory for the Imps, who had created the better chances in the second half and finished strongly after some attacking substitutions - including the matchwinner.
With significant player rotation for both sides, it is perhaps no surprise the match started slowly.
Brewers defender Julian Larsson earned himself a yellow card with just 10 minutes on the clock after bringing down Zane Okoro who was looking to break from a well-defended corner.
City created moments of danger early in the match, with an Erik Ring cross from the left going just too deep and an Okoro shot blocked behind for a corner by Jack Newall. Street saw an header go just over the bar.
Adam Jackson forced a save out of Kamil Dudek when a corner was played back into the area by Tendayi Darikwa. Ryley Towler's shot was blocked, then Jackson could not get enough power onto the shot to test Kamil Dudek.
Rob Street headed over from a Tom Bayliss cross, and Erik Ring lifted an effort over after a one-two with Frankie Okoronkwo.
The second half started with a sense of purpose from both sides, with Charlie Webster getting a shot away from distance just three minutes into the half which tested Zach Jeacock as the ball swerved in the air in front of him.
A nice set-piece move saw Bayliss loft the ball in for Tom Hamer, who was strong to win the ball and hold off his man before pulling it back for Ring - his effort lacked the power to test Dudek.
Moments later Dudek was again in action to halt Street's header on the stretch from a deep cross, then Dudek could only watch on as Street's far post header went just over.
Jeacock was called into his first meaningful action of the half after 61 minutes, getting down quickly to palm away a low shot from Dylan Williams.
Skubala made four changes with 20 minutes to go - sending on Draper, Conor McGrandles, Reeco Hackett and James Collins.
The home side's best chance of the game came after 74 minutes when the impressive Charlie Webster tried his luck with a free-kick from just outside the area. It clipped the top of the bar before bouncing over.
Hackett quickly got himself involved with a couple of good crosses - the second almost found Okoronkwo in a dangerous position.
It took a stunning save from Dudek to deny Collins after 81 minutes, standing up stall to stop City's No.9 who broke on a counter attack.
The decisive moment came after 89 minutes when Draper laid the ball off to House, who curled the ball magnificently into the far top corner.
Imps: Jeacock, Darikwa, Jackson, Towler, Bayliss (McGrandles, 71), Ring (Draper, 71), Street (Hackett, 71), Okoronkwo, Hamer, Varfolomeev (House, 84), Okoro (Collins, 71).
Subs: Pardington, Montsma, Bradley, Okewoye.
Burton Albion: Dudek, Armer, Moon, Delap (Sibbick, 46), Taroni, Williams, Websiter, Akoto (Lofthouse, 80), Newall (Shade, 58), Tavares, Larsson. Subs not used: Isted, Krubally, Thistleton
Attendance: 1,837
Referee: Oliver Mackie
