Johnson shot a 70 and Driscoll charged home with an 8-under-par 62 in the final round to tie at the end of regulation at 15-under 265.
The win was the sixth for Johnson on the PGA Tour. Driscoll was seeking his first win.
Bill Haas (65) and Paul Goydos (69) tied for third place at 14-under 266.
Justin Leonard, a three-time winner of the tournament, closed with a 69 go finish in a four-way tie for fifth place at 267.
Johnson had fired a course record-tying 60 in the third round.
"I know it's hard to back up a round like that, and I certainly didn't play my best today, but there was times there when I could have gone further south," Johnson said after the tournament. "Whether it was a one- or two-putt or a difficult shot or taking my medicine when I got myself in trouble, I kind of just remained patient and persevered."