Three-time winners Toulon appeared to be in line for a winning start as they opened up an 18-5 lead at half-time, thanks to tries from tries from Beka Gigashvili and Ben White.
An injury to replacement scrum-half Baptiste Serin, however, in the 56th minute shifted the balance and Exeter, who won the cup in 2020, closed the gap when Max Norey added to Harvey Skinner's first-half try.
Exeter then camped on the Toulon line and with 40 seconds left on the clock, the video referee confirmed that flanker Jacques Vermeulen had touched down for another Chiefs try.
Slade's conversion from five metres in flew between the posts to seal a thrilling victory.
It was a remarkable turnaround for the English side who were up against it in the early exchanges.
Exeter, though, got the first points on the board when Toulon's new English hooker Jack Singleton coughed the ball up for Chiefs fly-half Skinner who hit the gas to charge 50 metres to touch down.
The home side hit back almost immediately as Georgian prop Gigashvili burrowed over from close range.
Three minutes later the French struck again, White's long pass finding Gael Drean on the right wing. He brilliantly laid off to Charles Ollivon who was dragged down just a couple of metres from the line.
Scottish international White collected and nipped through for the try.
Enzo Herve was on target with a conversion and two penalties to make it 18-5 at the break.
The departure of Serin just two minutes after coming in changed the complexion of the game as Exeter sensed an opportunity.
Replacement prop Norey crashed over on the hour, Slade converting to bring it back to 18-12.
Toulon looked as though they had done enough to stave off the Chiefs before Vermeulen went over and Slade delivered the coup de grace.
Record five-time champions Toulouse kick off their campaign later on Saturday against Cardiff, while holders La Rochelle start their bid for a third straight title with a repeat of last season's final against Leinster on Sunday.
