Kara Mbodj opened the scoring for Genk from the penalty spot in the 29th minute before Rapid levelled through Terrence Boyd's header five minutes before the break.
Boyd was on target again in first-half stoppage time as Rapid pulled ahead, but Genk clawed their way back with Thomas Buffel's equaliser on the hour mark to salvage a draw.
Dynamo Kyiv's 2-0 away win over Thun, despite being reduced to 10 men, saw them move to seven points in the group standings to consolidate their grip on second place, now one point behind Genk following their slip-up in Austria.
Genk took the lead from the spot after Vienna goalkeeper Jan Novota was penalised for a foul on Benjamin De Cuelaer in the area.
Mbodji stepped up and fired the resulting penalty firmly into the goal's top right-hand corner to put the Belgian side ahead.
However, Rapid lived up to their name with two quickfire goals in five minutes at the end of the first half to go in at the break with a 2-1 lead.
Boyd was on target on both occasions, first heading in the rebound after Mario Sonnleitner's header clattered onto the post from a corner, before he claimed a second with an unstoppable header from Christopher Trimmel's well-flighted cross from the right.
Rapid looked on course to secure their first win in the group stages until the 60th minute when, from nowhere, Genk moved back on level terms.
Fabien Camus played in a ball from the left and Buffel got in front of his marker to flick the ball past Novota.
Brian Behrendt had a chance to wrap up a late win for Vienna in the 85th minute but he fired his shot straight at goalkeeper Laszlo Koteles as the tie ended locked at 2-2.