Jelle Vossen's 13th goal of the season just before half-time gave Genk the lead but Beveren struck back 18 minutes into the second half through Barak Badash.
Wednesday's result makes it three games since Genk last won in the Belgian First Division, while Beveren extended their unbeaten run to four matches.
In a first half of limited chances, Genk almost hit the front in just the fourth minute when French midfielder Julien Gorius struck the post from the edge of the penalty area.
The visitors' next chance on goal gave them the lead following Thomas Buffel's driving run into Beveren's penalty area.
Buffel's 39th-minute run was ended by two Beveren defenders but the ball remained loose and after Gorius' shot was blocked, Vossen was on hand to finish off the rebound.
The home team had two opportunities to equalise before the break with Israeli striker Badash going closest when his near-post strike was turned away for a corner by Genk goalkeeper Kristof Van Hout.
Genk almost regained the lead early in the second half but Beveren goalkeeper Colin Coosemans palmed away.
In the 63rd minute, Christoph Lepoint flew down the right before releasing Badash in behind Genk's defence and the Israeli side-footed the ball calmly past Van Hout.
As the second half wore on, Genk delivered the ball into dangerous areas on a number of occasions only for their forwards to fail to shoot.
Beveren almost made the visitors pay for their lack of cutting edge in the final stages when Badash got one-on-one with Van Hout but could not place his shot past Genk's goalkeeper.
Elsewhere, Zulte-Waregem needed an 88th-minute winner from Mahamadou Habibou to claim a 2-1 victory against OH Leuven.
Habibou gave the second-placed club the lead in the 69th minute when he finished off a fine pass from Mbaye Leye but Derick Ogbu's 81st-minute strike looked to have salvaged a point for OH Leuven.
Waregem had Habibou to thank seven minutes later as he scored after Franck Berrier's lead-up work to take the home team to within five points of leaders Anderlecht.
In other results, Lokeren thrashed Cercle Brugge 3-0, Mons defeated Sporting Charleroi 2-1 and Mechelen triumphed 1-0 over Gent.