The season-long loan signing of Drinkwater from Chelsea earlier this month was seen as a coup but, after Jay Rodriguez had given Burnley the lead, the ex-Leicester midfielder had an inadvertent hand in Will Grigg's leveller.
Burnley were then caught cold on the resumption, with Tom Flanagan and George Dobson scoring for the Sky Bet League One outfit against their Premier League opponents within five minutes of the restart.
While Sean Dyche retained only Dwight McNeil from the side that drew at Wolves at the weekend, this was a largely second-string Sunderland outfit, with Jack Ross making eight changes to his line-up.
Rodriguez marked his first start since re-signing with his hometown club with the opener, keeping his cool to calmly side-foot beyond Sunderland goalkeeper Lee Burge in the 11th minute.
Lennon and then Vydra rattled opposite posts within seconds of each other before Sunderland were gifted a route back into the tie when Drinkwater's attempt to cut out a Denver Hume pass fell straight into the path of an unmarked Grigg in the area.
The Northern Ireland striker seemed surprised to be handed the chance but he took it with aplomb, firing beyond Hart in the 35th minute to the delight of the buoyant Sunderland fans.
The visitors went ahead in the 47th minute as Hart was only able to parry a strike that looked like it may have deviated through the air from Lynden Gooch on the edge of the area, with Flanagan heading in the rebound from close range.
The sizeable travelling contingent were in dreamland in the 50th minute when a short corner was worked to Dobson on the edge of the area and the midfielder fired low through a group of bodies for his first Sunderland goal.