Playing in the showpiece match for the first time, Exeter were always ahead after scoring early tries but Racing, desperate to avoid a third final defeat, kept battling back and made it a one-point game going into the final 10 minutes, when Exeter were reduced to 14.
The English side held out though in a heroic goalline defence that will go down in club folklore, before flyhalf Joe Simmonds slotted a last-minute penalty to settle it.
It is an amazing success story for Exeter, who have gradually built themselves into a major force and who can complete a double when they face Wasps in next week's English Premiership final.
Exeter drew first blood in their usual way -- a catch and drive from a lineout that sent Luke Cowan-Dickie over in the eighth minute.
They were virtually camped on the Racing line from then until going over for their second try through number eight Sam Simmonds after 15.
Racing finally built some phases and a sharp pass by Finn Russell sent Simon Zebo over in the corner to bring them back into it at 14-5.
Argentine wing Juan Imhoff, whose late try secured Racing's semi-final victory over Saracens, then slipped through a rare gap for his sideG��s second try as Russell started to cause problems with a series of probing kicks.
Exeter, however, cashed in on another error to gain territory and, after repeated assaults, almost inevitably rumbled over via Harry Williams to send them in 21-12 ahead at the break.
Racing really needed to score first after the restart and did so when Zebo finished well for his second try, but the high-risk side of Russell's approach was exposed when Exeter's Jack Nowell intercepted a risky lofted pass deep in his own half and sent Henry Slade over to make it 28-17.
Yet again though Racing fought back as powerhouse hooker Camille Chat forced his way over and a Maxime Machenaud penalty cut the lead to a point with 15 minutes remaining.
Exeter then suffered what could have been a killer blow when replacement prop Tom Francis was sin-binned for a deliberate knock on, leaving them down to 14 for the last nine minutes.
However, incredible goalline defending kept out relentless attacks as Racing went through 19 phases before the English side managed a decisive turnover.
Exeter then earned a last-minute penalty that Joe Simmonds slotted, taking his goalkicking tally in the competition this season to a staggering 39 from 41 attempts.
Exeter's celebrations were briefly put on hold as referee Nigel Owens tried to establish if the clock had ticked into the red before the ball split the posts, finally deciding it had.
The only disappointment for a club that is at the heart of its community, was that no fans were there to see it, though remote celebrations are guaranteed to go long into the night.
Exeter's win follows Bristol's victory over Toulon in the Challenge Cup final on Saturday (NZT).