Bjorn surged into the lead with a brilliant nine-under 63 on Sunday to enter the clubhouse in the lead at 19-under. He missed a short birdie at the last that would have padded his advantage but nevertheless all he could do was wait.
Harrington, trying to redeem himself from last week's final-round collapse at the International Open, put together a solid effort to remain in contention while several players went low. At the par-five 15th, Harrington reached the green in two and walked away with a birdie to join Bjorn atop the leaderboard.
At the 18th, Harrington hit his approach to the opposite side of the green. With rain starting to fall, he left his birdie try short with a difficult putt left for par. Harrington collected himself and drained the 10-foot putt to finish regulation alongside Bjorn at 19-under-par 269.
'Coming down 18 I didn't feel nervous but I can't say I hit the purest of shots,' said Harrington. 'You've got to expect that under pressure.'
The players returned to the 18th tee to start the extra session. Bjorn, playing in his first playoff on the European Tour, knocked his second shot in a greenside bunker giving the advantage to Harrington.
Bjorn hit out of the sand and sent his ball eight feet past the hole. Harrington left his putt within three feet of the cup and after Bjorn failed to save par, the Irishman sank the putt for the victory.
'It would have been a real dent on my confidence if I hadn't won,' said Harrington. 'I thought I had licked it at the end of last season but last week I wondered if I was back to my old situation and not converting. It's a big confidence to get over the finish line and it's nice to say you are on the right track.'
Harrington, who has let several tournaments slip from his grasp in the final round throughout his career, played well enough to maintain control. His par save on the final hole in regulation ultimately gave him the chance to take the lead in the Order of Merit race.
Harrington struggled out of the gate with a bogey at the first. He responded with a birdie at the third and ran home an 18-footer at the par-three fifth to reach 16-under.
He kept things going with a birdie at the sixth and added a birdie at the ninth to make the turn at minus-18. Harrington then parred the next five holes before the clutch birdie at the 15th in his round of 68.
'I think I was playing very well today and stuck to my guns,' said Harrington. 'I hit a lot of good putts that didn't go in.'
Bjorn started with a pair of birdies from the third. He added birdies at the seventh and eighth before sinking a 12-foot putt at the ninth to go out in 31.
The Dane made it four in a row with a birdie at the 10th and continued his fine play with a pair for birdies starting at the 13th. Bjorn then birdied the 17th to secure a spot in the playoff.
'I played really well all through the round and even left a couple out there,' said Bjorn. 'Maybe I left too much to do coming into the weekend.'
Retief Goosen was also in contention late and failed to convert on a long birdie putt at the last to miss the extra session at 18-under-par 270. Goosen collected six birdies for a round of 66 to finish alone in third.
Niclas Fasth was one shot further back at 17-under-par 271. Fasth was on fire early with two birdies and an eagle over his first three holes to surge into the lead.
The Swede rolled in a long birdie at the fifth and birdied the sixth to reach 17-under. Fasth could not continue the rapid pace down the stretch, however, with two birdies and two bogeys the rest of the way for a 66.
'I gave it a real effort but always felt I needed two or three more birdies,' said Fasth. 'I put a little bit extra into it and came up short, but it was still an encouraging week.'
The English duo of Paul Casey and Justin Rose finished alongside Northern Ireland's Graeme McDowell in a tie for fifth at 16-under-par 272.
Darren Clarke, who struggled on the greens throughout the final round, was joined by Greg Owen and Paul Lawrie at 15-under-par 273.
Tiger Woods, who won this event the last two years, posted his best round of the tournament on Sunday. Woods hit a bunker shot to three feet for a birdie at the first and picked up another short birdie at the second.
The top player in the game hit his approach to 12 feet for a birdie at the 13th and birdied the par-five 17th for a round of 68.
'Overall, I'm very pleased with the way I played,' said Woods. 'I only had three bogeys the entire week so I didn't play bad, I played good. I just didn't get anything out of it on the greens.'