Pulling double duty in the weather-disrupted US PGA Tour event, Johnson used his formidable length off the tee and a sure putting stroke to build a big lead and held on down the stretch.
On a marathon final day he produced a third-round 64 and a final round of even par 71 for a 17-under par total of 267.
He led by as many as nine strokes before a few late miscues, but with the win will end Australian Jason Day's 47-week reign atop the rankings.
Johnson admitted the top spot wasn't something he dreamed of as a young golfer.
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Now that he's there, he said, "It's going to just push me to keep working harder and stay there."
Day could have maintained his position by finishing at least tied for third but closed with rounds of 75 and 71 to finish equal 64th on two-over 286.
"He deserves it," Day said of Johnson's rise -- which also took the American past second-ranked Rory McIlroy, who has been sidelined by a rib injury.
With number one on the line, Johnson made a blistering start on Sunday piling up seven birdies in his third round.
That included three straight birdies to cap the round -- a streak he continued when he returned less than 20 minutes later to open his final round with back-to-back birdies.
A birdie at the sixth moved him to 20-under for the tournament -- on pace to tie or break the 72-hole tournament record of 20-under 264 set by Lanny Wadkins in 1985.
But Johnson's run of 49 holes without a bogey ended at the ninth, and back-to-back bogeys at 15 and 16 saw his lead shrink further, although in the end that meant nothing.
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Belgian Thomas Pieters had eight birdies in a final-round 63 that pulled him into a share of second place on 12-under 272. He was tied with Scott Brown, who closed with a 67.
England's Olympic gold medallist Justin Rose had an eagle and four birdies in a three-under par 68 to head a group sharing fourth on 11-under 273.
That included PGA Tour rookie Wesley Bryan, who started the final round in second, five shots behind Johnson.
Bryan, a three-time winner on the Web.Com developmental tour last year, followed his third-round 63 with a one-over 72. Kevin Na shot a 70 and Charley Hoffman signed for a 67 to join the group on 11-under.
Johnson led by one after 36 holes, and was thrilled to triumph on a course where his six prior top-10 finishes included a playoff defeat in 2015 and one other runner-up finish.
"It feels good to finally win because I felt like I've been so close quite a few times, and it's one of my favorite courses to play on tour all year," Johnson said.