The former England fly-half missed out on 11 points with the boot and had handed over kicking duties to Matt Giteau before showing the magic remains when he split the posts from 30 metres with five minutes remaining.
Toulon, the holders, sit top of Pool Two following their win, with their earlier points in the contest having come from hooker Florian Fresia's try and a penalty each from Wilkinson and Giteau.
Director of rugby Bernard Laporte was satisfied with Wilkinson's move to hand over the kicking responsibility to his Australian team-mate.
"It was only Jonny's decision because he is the captain," said Laporte. "Three of the kicks he missed were from long range. Jonny is not a robot - he is a human being like everyone else.
Laporte said he was satisfied with Toulon's display in "a difficult match".
"We knew before the start that Exeter can play well but we also played well and the difference was good enough," he said.
"It was important for us to win one game outside of France but the competition is not over yet. We have a number of injuries after this game with Bakkies Botha, Bryan Habana and Craig Burden but that is the way with rugby."
The two sides meet again at Stade Felix Mayol next Saturday and Laporte is expecting another testing assignment.
Exeter could take heart from their performance which yielded two Gareth Steenson penalties and one from England Under-20 player Henry Slade.
"We have only been in the Premiership for four seasons and the Heineken Cup for a couple of seasons. Now we have gone toe to toe for 80 minutes with last season's European champions," said Exeter Chiefs head coach Rob Baxter.
"I don't think anyone would have been shocked if we had come through on top with a couple of penalties or whatever. You are always disappointed when you lose; I'm disappointed for the players who put so much into the game.
"Often when you put that much into the game you'd like to get something more than a losing bonus point. But the truth is we are not in any different a position than we were before the game and now we have to win two and one of those away from home.
"The pool is still alive for us; one is aware that it is a very tough group where nobody is going to dominate and win every game. So what we have to do now is dust ourselves down and go and do it all again next week."
Baxter paid tribute to 20-year-old Slade who had 60 minutes of action when he replaced the injured Steenson. Slade was one of the key performers for the England Under-20 side last season as they lifted a third successive Six Nations title.
He could be a starter in the return game with Toulon.
"I think a lot of teams would have been really dented when their number one fly-half went off so early in the game," added Baxter.
"I thought Henry stepped up and did really well and showed a real maturity.
"I think for a lad to come on in such a big game at that age, get that much game time and perform so well, is a credit to him. It bodes well for the future. We'll assess Steenson's muscular problem and make the fly-half call later in the week."