The reigning champions secured the required winning bonus point, but Joe Schmidt and his squad can only sit back and watch as Munster take to the field against Racing Metro at Thomond Park this afternoon knowing that a victory and four tries will see them snatch the final quarter-final spot.
"We went out with our mission to win the game and after that chase the bonus point and we got that, hopefully we did enough," said Heaslip. "Exeter really stuck it to us; it was a great game to watch and knackering to play but it's out of our hands now."
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Any hopes that both Irish provinces could take the spots available to the two best Pool runners-up were dashed by Montpellier, whose emphatic home victory over Toulon guaranteed them one of those places.
"They are two great teams going at each other," added Heaslip of the Munster-Racing game, "it's out of our hands. It will probably be the first time I'll watch a game of rugby in a long time."
Meanwhile, Ulster kept their hopes of a home quarter-final alive with a tense and scrappy 9-8 victory over Castres at Stade Pierre-Antoine – their first win in the competition in France. Ulster must wait on today's games before finding out if they will have a coveted home tie, as Toulouse and Saracens are also in contention.