West Ham boss David Moyes was denied a win in his 1000th game in management as Tomas Soucek's late own goal gifted Genk a 2-2 Europa League draw.
The Hammers thought they had won it when two second-half goals from Said Benrahma cancelled out Joseph Paintsil's early opener, only for Soucek to find the back of his own net in the 87th minute.
The Premier League side can qualify for the knockout stages with two games to spare if either Dinamo Zagreb and Rapid Vienna win the other Group H match later on Thursday.
Genk put an end to their losing streak at home, having lost their previous four home matches in all European competition – their longest ever such run in Europe.
John van den Brom's team took the lead inside four minutes when an incisive ball from Junya Ito found Paintsil, whose shot went in off the post after Alphonse Areola could only get a hand to it.
They never doubled their lead soon after as Areola was forced to save a Paul Onuachu header, before Ito sliced horribly wide after Paintsil pulled the ball back to him on a counter-attack.
Michael Antonio thought he had equalised a minute before half-time when he hit a shot under Genk goalkeeper Maarten Vandevoordt, only to see the ball cleared off the line by Patrik Hrosovsky.
West Ham started the second half brightly and were level after 59 minutes when Manuel Lanzini, Vladimir Coufal and Benrahma combined well for the former Brentford man to fire in.
Benrahma then got in down the left and slammed a shot past Vandevoordt again with just eight minutes to go, but a flick from Soucek from a dangerous cross looped over Areola and into the far corner to deny the Hammers all three points.