Exeter boss Rob Baxter admitted the chaos of the final few moments of his side's win over the Ospreys provided plenty of drama.
In the end the Chiefs progressed on points difference ahead of Bordeaux-Bègles, after Clermont failed to secure a losing bonus point in their loss to UBB.
Exeter will now travel to face Wasps in the Champions Cup quarter-finals in April.
"Qualifying from this pool looked very unlikely and anyone who put a bet on that outcome has probably made a fortune," Baxter told the EPC Rugby website.
"But we knew this was precious and we wanted something out of it. Admittedly we did not expect it to yield this reward but the players had the right attitude and have a tangible reward.
"It was pretty confusing at the end there, hearing Clermont turned down a kick to win the pool, but it has panned out for us even if we did not know whether to kick the ball out at the end or score another try.
"It is hard to express, but I am delighted for the players because now they have a memory that will last with them for many, many years to come.
"It is a fantastic achievement for us and we want to have as many big games as possible to look forward to towards the back end of the season.
"We are confident we can be in with a genuine shout of doing well because there will be an English club in the final and we are second in the Premiership. We can really enjoy it."