Toulouse stormed to a sixth final in Europe's premier competition courtesy of second-half tries from Yannick Jauzion and David Skrela in a 26-16 victory.
Heaslip, who started all three Tests for the Lions last summer, grabbed a late consolation for the Irish province but it proved to be too little, too late in the south of France.
"It wasn't for a lack of heart but Toulouse were the best team on the day and we can have no complaints," said Heaslip.
"They contested very well around the pitch and I don't think we controlled possession or territory very well.
"We gave them a couple of easy outs with the penalties in both 22's. They took their scores well. They were the better team, simple as."
Leinster were level at 9-9 until Jauzion and man-of-the-match Skrela crossed for quick-fire, second-half scores at the Stade Municipal.
But back-rower Heaslip reckons the Irish province were guilty of wasting their opportunities against the French giants.
"We were confident coming into the game and we didn't have any doubts over our ability on the pitch or the gameplan we were trying to play. Heaslip added:
"We had a couple of slips in concentration for the tries and they punished us for our mistakes in defence.
"But we weren't clincial ourselves. We had a couple of chances but that's the way rugby goes.