The defending champions trailed 12-6 at half-time, Brock James slotting four penalties to Jonathan Sexton's two - however an inspired team talk from the fly-half inspired a comeback.
Cian Healy scored the only try, converted by Sexton who also added a penalty, two minutes into the second half while Rob Kearney was on target with a drop goal.
Wesley Fofana had a try ruled out as Clermont offered a response, and while describing the contest as error-ridden, Cullen couldn't care less with another final booked.
"To be honest it was pretty intense out there. I think both teams made quite a few mistakes considering the quality of players out there," said Cullen.
"I think a lot of it was to do with the pressure that was there as well. It was one of those games that could have gone either way.
"It goes to the TMO - sometimes you get them, sometimes you don't. We got it there and at the end of the day that's the difference between the two teams."
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