He began the final day four strokes behind the pacesetting Levin and made a fast start before taking control with a spectacular finish that included a chip-in for birdie at the 16th that was lavishly praised by tournament host Jack Nicklaus.
"Boy, I hit it good today," Woods said after clinching his second PGA Tour victory this season in his final appearance before the June 14-17 US Open. "I hit the ball just as good as I have in years.
"I never really missed a shot today ... and I had the pace of the greens really nice today, where I struggled yesterday, and made a few putts."
Woods birdied four of the first seven holes and then three of the last four for a five-under-par 67 and a nine-under total of 279, drawing level with Nicklaus in second place for career victories on the US circuit.
Sam Snead leads the way with 82 PGA Tour wins.
"It's been a nice run since I've turned pro and to do it at age 36 is not too shabby," Woods said of matching his childhood idol Nicklaus.
"I've been very proud of what I've done so far, and I feel like I've got a lot of good years ahead of me."
Sitting beside Woods in the media center, Nicklaus graciously said: "If he's going to do it (win his 73rd title), which he was obviously going to, I'd like to see it happen here. That was good."
Sabbatini, who briefly led by two shots on the back nine, closed with a 72 to share second place at seven under with Argentina's Andres Romero, who rolled in a 13-foot birdie putt at the last for a 67.
Journeyman Levin, seeking a maiden PGA Tour victory, clung to a three-stroke lead at the turn but limped home to a 75 and a tie for fourth at five under with fellow American Daniel Summerhays (69).
On a breezy day of sunshine, the tournament turned on the astonishing flop shot Woods produced at the par-three 16th where he holed out from a poor downhill lie in greenside rough, 50 feet from the cup.
As huge roars erupted across the course, Woods unleashed his trademark uppercut fist pump to celebrate joining Sabbatini in a tie for the lead at eight under.
Playing in the group behind, Sabbatini bogeyed the 16th, after finding a bunker off the tee, to hand Woods a one-shot lead.
"That was the most unbelievable, gutsy shot I have ever seen," Nicklaus told CBS.
"Look at the position he was in. If he's short, the tournament is over. If he is long, the tournament is over. He put it in the hole."
Woods later commented of his miraculous shot: "It was one of the hardest ones I've pulled off. I was trying to get inside probably eight or 10 feet (from the cup).
"The lie was just a little bit marginal. For it to land as soft as it did was kind of a surprise because it was baked out and it was also downhill running away from me. It just fell in."
Woods, who wore his customary red shirt for the final round of a tournament, then tightened his grip by sinking a nine-foot birdie putt on the 18th, sparking further roars from the huge galleries watching from below the clubhouse.
"Thanks Jack, I appreciate it," Woods said after being congratulated by the tournament host on his fifth win in the Memorial, before walking away to sign his card.
After battling a fever for much of the week at Muirfield Village, Woods will climb to number four when the world rankings are issued on Monday.