North Carolina-born Love, 51, secured his 21st PGA Tour title, and his third in Greensboro, in magnificent style, recovering from a bogey at the first hole to card six-under-par 64 at Sedgefield Country Club.
More than two years after spinal fusion back surgery, and five months after foot surgery, the 2016 U.S. Ryder Cup captain finished at 17-under 263, one stroke ahead of overnight leader Jason Gore (69), who missed a long birdie chance at the last that would have forced a playoff.
"Any victory now when you're over 50 is really sweet," Love, who earned a berth in the lucrative FedExCup playoffs with the win, told reporters. "It's hard to believe."
Woods, who has not won on the PGA Tour since the 2013 WGC-Bridgestone Invitational, began the day two strokes off the lead and needing a win to guarantee a FedExCup playoffs berth but was not able to convert a birdie opportunity until the ninth hole.
"I gave myself a chance. I had all the opportunities in the world to do it and didn't get it done," said Woods.
Love ended a losing streak on tour that stretched back to 2008 by making eagles at both par-fives, and adding four birdies to go with two bogeys. And he made his move while ignoring leaderboards.
"I did a really good job (of) not looking at leaderboards. When I walked off and saw I had a two-shot lead I was shocked," he said. "I've been saying all year if I putt well enough to win on the (over-50s) Champions Tour I can win on this tour.
Love, who arrived in Greensboro ranked 613th in the world, gave Woods some credit for his victory.
"Everybody thinks I'm just being nice trying to hang around with Tiger and help him out, and I am, but part of it is I want to learn.
"He's really helped me a lot with prepping for tournaments (and) he gave me a little nudge with my putting back at the PGA."
Love is now the oldest living winner on tour. Sam Snead was 52 when he won the 1965 Greater Greensboro Open at Sedgefield, and Art Wall was 51 when he won the 1975 Greater Milwaukee Open.
(Editing by Frank Pingue)
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