The Wales international came off the bench to snatch a crucial away goal for the visitors, cancelling out Gary Mackay-Steven's first-half penalty and giving the Clarets the advantage heading into next week's return fixture.
Shortly before the twist 10 minutes from time, replacement keeper Anders Lindegaard produced an impressive save to keep out Mackay-Steven from close range in a well-contested all-British derby clash.
Aberdeen allowed their opponents to take control of possession in the first half, but it was the hosts who enjoyed the better of the rare openings.
Niall McGinn sent an angled shot wide of the target after his initial cross, crying out for a touch, went right the way through the box and back to him.
Five minutes later German referee Daniel Siebert pointed to the penalty spot after James Tarkowski made slight contact with Sam Cosgrove inside the area, allowing Mackay-Steven - on penalty duty following the recent exit of Adam Rooney - to tuck away from 12 yards.
The match became more open for the final 30 minutes, but it was Aberdeen who were looking the more likely to score as Cosgrove drove the ball into the hands of Lindegaard, and the replacement keeper then pushed Mackay-Steven's header around the post from point-blank range.
That would prove to be a crucial save as, after Gudmundsson and Vokes were wayward with their attempts, the latter found a way through to temporarily silence the home crowd.
The two sides will meet again next Thursday evening at Turf Moor, with the winners to face Turkish side Istanbul Basaksehir for a place in the competition proper.
ABERDEEN (4-3-3): Lewis; Logan, Devlin, McKenna, Hoban; Ferguson (Gleeson 57'), Shinnie, Ball; Mackay-Steven, Cosgrove (May 76'), McGinn (Wright 79')
BURNLEY (4-5-1): Pope (Lindegaard 14'); Lowton, Tarkowski, Mee, Ward; Hendrick (Vokes 57'), Cook, Westwood, Gudmundsson, Lennon; Wood