But the French side put daylight between themselves and Leinster in the second half, winning 29-14 to book their place in the European Cup semi-finals.
And Heaslip, who appeared six times on Tour for the Lions in Australia last summer, believes his side were never able to get their game going.
"It's hard to take, it's never nice losing, we knew the challenge that we had coming over here," he said.
"We had to be accurate and had to be 100 per cent on our lineouts, scrums, execution and focus.
"We simply weren't and they capitalised on it, they controlled the game very well and we just weren't accurate enough. They deserved to win it."