Karlsson lost to Westwood, who became the first European to win the long-established tournament, on the fourth hole of sudden-death.
The Englishman claimed his second US PGA Tour victory, but his first since the Zurich Classic of New Orleans in 1998.
“You try to do the right thing all the time,” Westwood said. “It doesn’t always work for you. I’ve been in contention a lot, especially this year, and I suppose I got a break today with other people’s misfortune but made the most of it and took a chance.”
American Robert Garrigus blew a three-stroke lead on the final hole of regulation with a triple bogey. He then bogeyed the first playoff hole to drop out of the race.
Westwood and Karlsson both went par-par-bogey until they returned again to the 18th at TPC Southwind.
Westwood landed his approach shot six feet from the pin, while Karlsson left himself a 43-footer for birdie. The Swede left that short while Westwood drained his birdie putt for the victory.
The trio all finished 72 holes on 270, Westwood after a 68, Karlsson a 69 and Garrigus a 71.
Westwood started the day three shots off the pace, but a burst of three birdies in a row from the second saw him take the lead.
Westwood drained a 30-footer at the second and holed out from nearly 34 feet on the third. At the fourth he rolled in a 19-footer.
He gave back a shot at 17 and with Garrigus up by three was ready to leave the course after his round until he was told to hang around.
That was when Garrigus, who had never led a US tour event on the final day, had put his tee shot at 18 into the lake alongside the fairway. He took a drop then hit his next shot into the trees. He ended up two-putting for a triple bogey to join the playoff.
Garrigus said he thought he had a two-stroke lead going into the final hole rather than three.
“It’s little things to win. I’ve got to learn that, and next time I’m in that position I’m going to do it,” Garrigus said.
Returning to 18, Garrigus stayed away from the lake, but his drive landed behind a tree so that he could only punch out into the fairway.
Westwood and Karlsson both safely parred, but Garrigus couldn’t make his 13-footer to save par and was out.
Karlsson had a chance to win at the third playoff hole with a par putt from inside six feet, but missed.
“I didn’t hit a good putt. You can’t take any chances with too many good players,” said Karlsson, who won his ninth European Tour victory at Qatar earlier this year but remains in search of a first US tour title.
“It’s amazing how things pan out,” said Westwood, who was playing on a sponsor’s exemption that allowed him to hone his game for the year’s second major, which starts Thursday at Pebble Beach, California.
“I like being competitive before a major championship, to be competitive with the tournament and boost my confidence,” Westwood said.