The Chiefs charged into a 14-point lead less than midway through the first half of a pulsating final at Ashton Gate, Luke Cowan-Dickie and Sam Simmonds going over.
Harry Williams added a third try just before half-time after Simon Zebo and Juan Imhoff crossed at the other end in a breathless contest in Bristol.
Zebo finished for a second time, but a poor pass from Finn Russell, who laid on both scores for the Irish winger, resulted in a Henry Slade try that gave Exeter breathing space.
Racing roared back again, Camille Chat touching down and Maxime Machenaud, who hit the post with one conversion attempt, reducing Exeter's lead to only a point with a penalty.
Exeter was a man down after Tomas Francis's yellow card eight minutes from time, but defended for its life and Joe Simmonds took his tally to 11 points with the last kick of the game.
The Chiefs can complete an incredible double if they win the Premiership next weekend a decade after they were promoted to the top flight for the first time, while Racing has now been the runner-up in three of the past four years as its wait to win the trophy goes on.
Exeter made a storming start with the forwards dominating, Cowan-Dickie crashing over when Racing were unable to stop a driving maul eight minutes in and Sam Simmonds crossing from short range.
Joe Simmonds converted both scores, but after looking shellshocked early on, the Top 14 heavyweight came to life and Russell fed Zebo for his first score in the corner.
The clinical Imhoff caught Exeter out with a show-and-go, but Rob Baxter's side led 21-12 at half-time after Williams scored just before the break follow a sustained spell of pressure.
Racing responded after the interval, Zebo capitalising on slack defending by bouncing off a tackle and spinning to touch down, having taken another pass from Russell, but Machenaud's conversion attempt hit the upright.
Slade was somewhat fortunate not to make contact with Imhoff's head in a tackle and he capitalised by going under the posts after Russell's poor pass was intercepted by Jack Nowell.
Racing hit back again, powerful Chat ploughing over following a driving maul and Machenaud adding the extras, then cutting the gap to a point with a penalty after 64 minutes.
Francis was sin-binned for a deliberate knock-on, but Exeter held on and Joe Simmonds had the final say to make it an English European double after Bristol Bears beat Toulon in the Challenge Cup final on Saturday (AEDT).