The Clarets, who scored late on in the first leg to secure a 1-1 draw at Pittodrie, began brightly in the return encounter and took the lead after just six minutes at Turf Moor.
The visitors, however, grew into the game and drew level in spectacular fashion just before the half-hour mark, 18-year-old Lewis Ferguson launching himself into an acrobatic overhead kick to find the top corner.
Both sides weighed in with dangerous chances during the second half, with Lewis making two fantastic saves to keep the Scottish Premiership side on level terms, but the final whistle could not separate either, which meant extra time.
The game continued at a relentless pace as both sides gave their all to remain in Europe, but Burnley would be the ones to nudge themselves in front as Charlie Taylor's cross on 101 minutes was headed home by Jack Cork.
With away goals counting at this stage of the competition, an unanswered Aberdeen strike would see the Scots progress through to the next round, but Sean Dyche's side were given the opportunity to effectively kill off the tie when Johann Berg Gudmundsson's cross hit Scott McKenna's arm in the box.
The referee rightly pointed to the spot and Ashley Barnes made no mistake, firing his penalty into the bottom right-hand corner on 114 minutes to extend the Lancastrians' lead.
Burnley saw the game out comfortably after their third goal and the final whistle confirmed their place in the third qualifying round, where they will meet Turkish side Istanbul Basaksehir.
BURNLEY (4-4-2): Lindegaard; Lowton, Tarkowski, Mee, Ward (Taylor 91'); Lennon (McNeil 87'), Cork, Westwood (Hendrick 106'), Gudmundsson; Vokes, Wood (Barnes 46')
ABERDEEN (4-3-3): Lewis; Logan, McKenna, Devlin, Considine; Shinnie, Hoban (Ball 88'), Ferguson (Forrester 106'); Mackay-Steven, Cosgrove (May 100'), McGinn (Wright 81')