Edin Dzeko's wonderful double and Frank Lampard's fourth goal in three games sealed a 4-2 victory for City
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Hull had to settle for a goalless draw on their competitive European debut as Tom Huddlestone missed a penalty and a point-blank follow-up against Slovakia's AS Trencin.
The Tigers, having qualified for a first ever campaign at Europa League level by finishing runners-up in the FA Cup, should have come away from their third qualifying round first leg at Zilina's Stadion pod Dubnom with victory but will now need a result in East Yorkshire next Thursday.
An hour had gone when Shane Long was felled in the area but Huddlestone - who missed in the shootout when Spurs lost to Basel in the 2013 competition - produced a dreadful double miss.
His penalty was comfortably saved by Milos Volesak but it was the secondary attempt, grazing the bar from a couple of yards with the goal yawning, that will haunt the midfielder.
Sheffield United have accepted a bid from Hull for defender Harry Maguire.
The Sky Bet League One club revealed they have "reluctantly" accepted the Tigers' new offer after they met their valuation of the centre-back and he and his representatives are now set to open talks over a move to the KC Stadium.