Rees says they felt there was "no drama" trailing 14-7 at half-time.
But defeat left Ospreys - and Welsh regional rugby - without a top-tier knockout place since 2010.
And as they trail in the Pro12, captain Alun Wyn Jones admitted Ospreys face a tough task to make the 2016-17 tournament.
They are currently seventh in the Pro12 table, four points behind Ulster, who occupy the sixth and final Champions Cup qualification place.
Wales lock Jones said: "We have to focus now on a tough Six Nations period because we are currently not in a position to play in this competition this year."
Hanno Dirksen crossed in the opening period against Exeter and Ospreys were awarded a penalty try that gave them late hope.
But two tries apiece for Thomas Waldrom and James Short in addition to Kai Horstmann's opener saw off the visitors.
Rees said Ospreys felt "pure disappointment" at failing to to take their opportunity to progress against the second-placed team in the Aviva Premiership.
"We spoke at half-time that there was no drama, we could build our way into the game and create some pressure without overdoing it against Exeter," said Rees.
"And I think we did that at the start of the second half, even at the end when we were 26-17 down. If we can negate their work in the 22 you'll get another shot and that potentially takes you through.
"We knew all that. Al [Jones] was getting messages - he was pretty clear, but it's redundant anyway."
Jones said: "It wasn't the ideal start. Two quick tries put us on the back foot, but in the same breath it was probably the best time to have them because we've got another 70 minutes to go.
"Unfortunately we probably chased a bit too much.
"We brought ourselves back into contention, but the game got away from us and and we had too much to do."
Ospreys went into the final round top of Pool Two after earning two bonus points in defeats at Clermont Auvergne and Bordeaux-Begles and beating both French Top 14 teams and Exeter at their Liberty Stadium home.
Jones said: "There's a lot of emotion involved in those performances we've had. We've probably lived off emotion and performed off emotion, but you can't really afford to do that in this competition.
"I remember being away in London Irish in the same competition. We went out there with a team that was probably on paper far superior and had the same result, so I think Gruff has nailed it on the head with the word 'devastated'.
"I said to the guys at the end that none of you in here [the media] said we would be in contention to get out of the pool and it's true. I remember doing the interview at the start of the season and somebody laughed, which is fine.
"And it's not about defying the odds or proving other people wrong, it's about working for each other and trying to do something different.
"We've made a point about all the big names that have gone through this jersey and not got ourselves in the position that we have.
"Ultimately we've faltered again and we haven't done that [gone through], but I can say credit to the team and we're proud to have come so close."
Ospreys backs coach Gruff Rees says they were "emotionally devastated" at missing out on a European Champions Cup quarter-final by losing to Exeter.