The Turkish Airlines Open champion, who finished with a birdie for a round of 67, ended Saturday on 11 under par, a shot clear of Argentina's Eugenio Chacarra.
Australia's Danny List has faded from contention. Going into the day just four shots off the lead, he carded an unhelpful 70 to fall six shots behind.
Couvra found himself two off the pace after 11 holes in Tuscany but clawed his way back to the summit after his partner, Spain's Angel Ayora, stuttered to a 70.
"I'm really proud about my 18th, because I was really tired on the last few holes. It's quite difficult with the heat," the Frenchman said.
"The course is so tough, so we needed to be very concentrated. I'm happy about the 18th.
"There's a lot of great players here, so you have to keep going. (The lead is) only one shot. I'll try to do my best on the last 18, and that's it.
"It's going to be a great experience because it's my first time being in this position. I'm really proud to be in this position and I hope to be there tomorrow."
Scotland's Calum Hill moved to within three shots of the lead after a third-round 64, with England's Alex Fitzpatrick alongside him on eight under.
Another Australian, David Micheluzzi, who started six shots adrift of the lead, hit a 71 to slip nine in arrears.
