The Clarets were held to a goalless draw by their Turkish opponents, replicating last week’s away leg stalemate, but substitute Cork rifled a 97th-minute winner at Turf Moor as a 1-0 win kept the club’s continental adventure going.
The once-capped England midfielder netted at a similar stage in the previous round to break the deadlock against Aberdeen and he will get the chance to extend his unlikely scoring streak against Olympiacos next week.
Joe Hart had a night to remember on his home debut, making a series of strong saves in an extended duel with Edin Visca, and has yet to concede after three outings for his new club.
Dyche’s efforts in steering Burnley into Europe earned him considerable praise but he has made it clear the competition must come a clear second behind the Premier League.
As such there were seven changes in his starting XI, with a debut for £15m defender Ben Gibson the most notable.
But it was the recalled Ashley Barnes who was most eager to impress. He barrelled into challenges, chased long balls and almost scrapped his way to an early opener after upending Alexandru Epureanu and finding the side netting.
Phil Bardsley’s long-range drive worked Mert Gunok but Burnley’s possession quickly lost focus. Basaksehir only woke up in the 10 minutes before the break and could easily have taken the lead. Visca lashed one sighter over the bar but tested Hart’s reactions with his next two attempts. Kerim Frei was next to try his luck, but Hart dropped to his right to tip wide.
The tie was there to be won and both sides had half-chances, Sam Vokes from a corner and Visca once again failing to get the better of Hart with a skiddy effort.
But, just as he had done against Aberdeen, Cork made the difference.