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.
What does it mean? Hammers made to wait
There was to be no victory for Moyes in his landmark game, but the Hammers are still on the cusp of qualification for the knockout stage after four games.
They can be relatively pleased with going to a strong Genk side and getting a point, which was only West Ham’s second trip to Belgium in all European competition after a 1-0 win over KAA Gent back in September 1964 in the Cup Winners’ Cup first round, with Ron Boyce scoring the winner.
Where has this Genk been?
After an impressive 1-0 win at Rapid Vienna, Genk’s Europa League campaign stumbled badly with back-to-back 3-0 defeats against Dinamo Zagreb and West Ham.
The Belgians were a match for their English opponents here, though, having 14 shots with five on target, and very nearly equal possession (49.6 per cent). Play like this in their final two games and Genk will fancy their chances of still making it through.
Waste Ham
Even if a draw is not the end of the world, Moyes will still feel a sense of frustration after seeing his side waste numerous chances to win the game.
West Ham had 15 shots in the game, with 13 of those being taken from inside the penalty area and seven hitting the target, but only finding the net twice.
What’s next?
West Ham host Liverpool in the Premier League on Sunday, while Genk are also at home on Sunday as they face Cercle Brugge in the Belgian First Division A.