The Belgian side travelled to London having lost four of its previous five matches across all competitions and West Ham piled on the misery to go six points clear at the top of the group.
Genk looked bright during the early exchanges but West Ham grew into the occasion and went ahead on the stroke of half-time, Craig Dawson's looping header finding the net.
West Ham put the game beyond Genk with a quickfire double just before the hour, Issa Diop and then Jarrod Bowen scoring 72 seconds apart to leave David Moyes' men on the brink of qualification for the knockout phase.
West Ham briefly appeared to have fallen behind early on as Theo Bongonda's diving header found the net, but the host was let off the hook as Junya Ito strayed offside in the build-up.
Maarten Vandevoordt was alert to prevent Jhon Lucumi putting into his own goal from an Aaron Cresswell cross just before half-time, though the Hammers did get the breakthrough from the resulting corner.
Dawson beat two markers in the air to meet Cresswell's delivery and Bryan Heynen could not prevent the header finding the top-left corner.
Another Cresswell set-piece brought West Ham's second goal, Diop seeing his header go in off the crossbar in the 57th minute.
A little more than a minute later, the Premier League side wrapped up the win as Manuel Lanzini found Bowen, the winger ran at the heart of the Genk defence and his shot squirmed past Vandevoordt.