The Belgian visitors took a 10th-minute lead through Jean-Paul Boetius' header but Celta hit back in emphatic fashion and were ahead inside seven minutes thanks to goals from Pione Sisto and Iago Aspas.
John Guidetti added a third before the break as Celta threatened to sweep aside their opponents, but veteran Genk playmaker Thomas Buffel pulled a goal back to leave the tie evenly balanced heading into next week's second leg.
Celta are looking to reach the semi-finals of European competition for the first time in their history but their hopes suffered an early setback as Genk took the lead.
Leandro Trossard did well to collect an over-hit cross from the right, create space for himself and then deliver a perfect ball back into the box that Boetius headed past Sergio Alvarez.
Having seen their side beaten 2-0 by Eibar in their last home outing, Celta fans might have feared the worst at that point, but Eduardo Berizzo's men roared back in style.
They equalised four minutes later with a superbly-worked team goal that was clinically finished off by Sisto, who picked up a clever flick by Guidetti before curling a shot past Matthew Ryan and into the roof of the net.
Read the full article on eurosport.co.uk: Aspas on target as Celta edge five-goal thriller with Genk