Matthieu Pavon takes historic PGA Tour triumph at Torrey Pines
Up by two strokes with two holes to play, Pavon missed a three-foot par putt at the 17th to see his advantage cut to one.
He was in a fairway bunker off the tee at the par-five 18th, where his second shot left him in the dense left rough.
But he muscled a mighty shot to the green and avoided a playoff with a clutch seven and a half foot putt, carding a three-under par 69 for a 13-under par total of 275.
Denmark's Nicolai Hojgaard carded a one-under par 70 to finish second on 276.
The victory on the demanding South Course at Torrey Pines - a two-time US Open venue - comes after Pavon captured his first DP World Tour title - in his 185th start - at the Spanish Open in October.
He earned his US PGA Tour card with a top-10 finish in the DP World Tour standings.
"I have no words. It's amazing," Pavon said, his voice shaking in a post-round television interview. "It has been a long journey. I waited seven years on the main tour to win back in Europe and finally got a win in America. It's just tremendous."
Pavon shook off an opening bogey with four front-nine birdies, his two-foot birdie at the ninth pulling him into a tie for the lead with Germany's Stephan Jaeger.
Jaeger, also chasing a first PGA Tour win, came into the final round with a one-stroke lead and pushed it to three with birdies at the second and third holes.
But three back-nine bogeys proved too much to overcome and Jaeger finished tied for third on 11-under with Americans Jake Knapp and Nate Lashley.
Pavon played the final nine in even par, but it wasn't all smooth sailing.
The Frenchman looked to be in trouble at the 14th, where his second shot went through the green down the steep bank behind the putting surface.
Luckily, it stopped before rolling out of bounds and his shot up to the putting surface rolled within a foot of the cup.
He was in a greenside bunker off the tee at the par-three 16th but drilled a 23-foot putt to save par and keep his lead at two strokes.
The guts to finish
After the bogey at 17, Pavon watched Hojgaard belt it down the fairway at 18 and thought "it's going to be spicy now."
After his second shot left him in dense rough, his caddie advised caution, but Pavon opted to go for the green.
"I was so pumped at that time, I know I had the energy to lift that ball up on the green," he said. "That ball came out like a butterfly and it really fed the slope on the green ... That was the right time to prove I have the guts to finish that tournament and I did it so I'm so happy about that last hole."
Hojgaard finished with five birdies and three bogeys in his two-under par 70, keeping the pressure on Pavon by reaching the 18th green in two and giving himself a two-foot birdie putt.
"We were all grinding our heart out," Hojgaard said. "(Pavon) made good up-and-downs on the back nine and gave me a chance on 17, but then he pulled some magic on 18, which you've got to do if you want to win.
"I gave it my best to put a bit of pressure on him today, but he got one on me. Happy for him."