The English champions came out on top in a lively clash at Murrayfield to successfully retain their continental title.
Clermont recovered from 12-0 down and, even after Owen Farrell edged Sarries eight clear, refused to give up in a thrilling Edinburgh final.
Ashton opened the scoring when crossing over 12 minutes in, taking him to 37 in European competition overall - clear of Vincent Clerc as the all-time top scorer.
Speaking after the match, Ashton told BBC Sport: "It's pretty hard to put that into words right now. We've worked so hard for the past five or six years.
"It's such a pleasure to be with this group. It's so hard to be in these finals, so to win two is a pleasure. To become top try-scorer is just the icing on the cake."
Saracens are the fourth side to retain the European Cup, joining Leicester Tigers, Leinster and Toulon in doing so.