Stade Francais booked a second successive Amlin Challenge Cup semi-final place as Samoan centre Paul Williams snatched a late try to see them edge past Exeter.
They will be away to either Toulon or Harlequins on April 27, and will be thankful to be through.
Exeter's hopes of European silverware go on hold as the Devon side now have to turn their attention to securing a Heineken Cup place for next season and a push for an Aviva Premiership Rugby semi-final slot.
The teams were tied at 17-17 heading into the very late stages, after four penalties by Ignacio Mieres and a try from Sireli Naqelevuki for Exeter. Extra-time seemed likely, but Williams pounced for the decisive try in the 79th minute to shatter the visitors.
Exeter were the team who had applied the early pressure, and they would have had an early penalty in front of the posts but Irish referee Peter Fitzgibbon allowed the play to continue looking for an advantage which was never really there. The ball went to touch.
The Chiefs continued to pin the French side back into their own half, with scrum-half Haydn Thomas making two bursts into the home 22 which threatened to break the early deadlock.
The first points arrived in the 21st minute when Stade Francais were penalised for going over the top and Argentinian fly-half Mieres slotted a 15-metre penalty between the uprights.
Four minutes later, after more Chiefs pressure, the Paris side were again penalised at the scrum but this time Mieres saw his 45-metre attempt sail past the left post.
Stade Francais were reduced to 14 men when their own Argentinian fly-half, Felipe Contepomi, was yellow-carded for fighting on the floor with Chiefs hooker Chris Whitehead and throwing a punch.
Mieres ignores the jeers of the French supporters to put the Chiefs further ahead with his second penalty, only for scrum-half Julien Dupuy to respond with a 40-metre kick after a high tackle.
Just after the half-hour mark, Mieres made a break before number eight Richard Baxter burrowed his way over the line. The referee referred the effort to the television match official, Scotland's Jim Yuille, who ruled it was not a try.
Minutes later, Mieres landed his third penalty before Stade Francais, now restored to full strength, hit back. Dupuy kicked his second penalty on the stroke of half-time, leaving Exeter 9-6 in front.
The Parisians started the second half strongly and Contepomi put a neat grubber kick through for full-back Djibril Camara who collected the awkward bouncing ball and dived over the line to put the home side ahead.
Dupuy was off target with his conversion attempt from the left touchline but then he added penalties either side of one from Mieres, meaning the Chiefs were five points adrift going into the final quarter.
That lead was wiped out though as Stade Francais prop Rabah Slimani received treatment. The game continued, and replacement Naqelevuki shrugged off a tackle to power over the line to level the scores in the 67th minute.
With extra-time beckoning, and Exeter pressing, Mieres sliced a drop-goal attempt off target. Stade Francais soon made a break led by Dupuy that put in Williams for the killer try.