Seven months after his first victory at La Cantera Golf Club, Johnson beat James Driscoll with a 10-foot birdie putt on the first extra hole after a final-round shootout in which seven players stood within a stroke of the lead with four holes left.
"I feel very lucky," said Johnson, who followed his third-round 60 with a 70. The 2007 Masters champion earned $1,098,000 for his sixth career PGA Tour victory.
Driscoll, eight strokes behind at the start of the round, shot a 62 to match Johnson at 15-under 265.
Johnson vaulted to the top of the FedEx Cup standings, passing Geoff Ogilvy and Phil Mickelson the only other two-time winners on the PGA Tour this season. Johnson also won the Sony Open in January.
Paul Goydos had a one-stroke lead with two holes to play, but closed with two bogeys for a 69, leaving him a stroke back along with Bill Haas (65), who birdied five of six holes on Nos. 11 through 16 but missed a 6-foot putt on the par-3 17th.
Australia's Marc Leishman (68), Sweden's Fredrik Jacobson (67) and three-time champion Justin Leonard (69) finished at 13 under.
Driscoll was an afterthought at 5 under when the final round began, eight strokes behind Johnson and his group that included Goydos and Leonard. After wrapping up his 62, he had to wait more than an hour for Johnson to finish his round.
It was worth the wait, but it didn't last long.
Johnson quickly won it with a 6-iron approach to set up his winning putt on the par-4 18th. Driscoll's birdie attempt from 27 feet rolled 2 feet wide.
"You want to put the pressure on your opponent. There's no doubt about it," Johnson said. "It's not like he hit a bad shot. He hit 20-plus feet. He hit a good putt. So it just kind of went my way."
Driscoll was in position for the biggest comeback in the history of the Texas Open, which dates to 1922 and is the third-oldest event on the PGA Tour. Instead, Driscoll settled for only his third top-10 finish and finish since a runner-up effort in the 2005 Zurich Classic of New Orleans.
"When you get that close to a win and that close to going to Augusta and going to Hawaii for the first tournament of the year, it's a little disappointing," said Driscoll, who has conditional status on the tour after finishing 141st on the money list last year.
Goydos briefly topped a crowded leaderboard with a 13-foot birdie putt on the par-4 16th before falling back on the last two holes, putting the 44-year-old and sympathetic favorite out of the running.
Goydos was playing four months to the day his former wife died. His final flaw, muffing his chip shot to a measly 7 feet on 18, denied him a spot in the playoff.
Goydos, who started the round two strokes back, led after 36 and was eyeing his first tour victory since 2007.
"For 70 holes I did really good," Goydos said. "Didn't hit a good putt on 17 and didn't hit a good shot on 18. And guys who play like that will do it."
Johnson is the first repeat winner at La Cantera since Leonard in 2000 and 2001.
Next year, the tournament will move to a Greg Norman-designed TPC course.