Converted tries by winger Fergus McFadden and scrum half Isaac Boss put Leinster 20-6 ahead at the break but Wasps then came alive and Matt Mullan went over after 61 minutes with Andy Goode converting from the left touchline.
Number eight Nathan Hughes then dived over the maul to touch down and Goode's trusty boot levelled the score with 10 minutes remaining at the Ricoh Arena.
Ian Madigan missed a late penalty for Leinster while Goode had a chance with a last-gasp drop goal.
A draw was enough for Leinster to book a last-eight place and leave Wasps sweating on being one of the three best runners-up having edged out Harlequins to second spot in the group.
"We probably should have won that game. We didn't get it right in the first half," Wasps back row James Haskell said.
"Every time we went into their half we came away with points, but we only went into their half three times.
"They looked much the better side in the first half, but we stepped it up in the second half."
In a Pool Five decider later on Saturday Northampton Saints are at home to Racing Metro, although they are both guaranteed a quarter-final place after the Wasps result.
Both have 19 points, already enough to ensure one of the best runners-up berths.