The Top 14 side led 32-8 after 55 minutes of the game at Stade Marcel-Michelin, Morgan Parra kicking his fourth penalty to put his side in complete control.
The scrum-half had also been on target with both conversion attempts after first-half tries from Fritz Lee and Damian Penaud, the pair touching down in an impressive four-minute spell for Clermont.
Yet Quins, which saw Mike Brown ride a tackle to reach the line just before the interval, rallied impressively to make for a nervy finish.
Chris Robshaw forced his way over and after adding the extras, Marcus Smith used his boot to good effect again with a grubber that bounced kindly for James Lang to finish.
Alex Dombrandt closed the gap further as Quins capitalised on a lineout opportunity to touch down again, yet the Premiership club ran out of time to complete its recovery mission.
After surviving a late scare, Clermont can now look forward to taking on La Rochelle, which defeated Sale Sharks in the first semi-final, for the trophy at St James' Park on 10 May.