The score was 12-12, with four penalties apiece, after 80 minutes but Ian Madigan's pass was intercepted by Bryan Habana and the South African raced away to score what would prove to be the decisive try.
Sean O'Brien plunged over in the second half of extra time to ensure a grandstand finale but the Toulon defence held firm for a famous triumph.
Leinster will now turn their attention to the GUINNESS PRO12 where they currently lie fifth, eight points outside the play-offs, but Heaslip refused to blame Madigan's intercepted pass for their European exit.
"It's not down to one player. It's down to the group, the collective to be honest. At different times, all players made mistakes in the game, on both sides," said Heaslip.
"So it can't be down to one moment. We left opportunities out there in the first 40. We had position and momentum to get more return than we did. In the third quarter, they had a purple patch and we stood our ground.
"It felt like a boxing match, a slugfest, punishing each other for making mistakes, especially when one of us got to the other side. It showed there was nothing between the sides.
"We always have an inner belief, an inner confidence, that if you do your job and have confidence in mates that they can do their job, we can stress teams and get points.
"We spoke quickly afterwards of having that confidence. We have a massive challenge in the league.
"We knew we had to produce some of our best rugby and do all the people who followed us proud. Hopefully we did that in our effort and our actions."
Follow us on Facebook, join the conversation on Twitter, sign up to our YouTube channel for extensive match highlights and sign up for our newsletter for regular updates on the GUINNESS PRO12
63x46.png){kind=logo}