Bjorn two-putted for par on the first extra hole, No. 18, while Howell got up and down for par from left of the green. Davis fell out of the extra session after just one hole as he was unable to save par.
Bjorn and Howell returned to the 18th tee for the second playoff hole. Howell missed the green right in a bunker. Bjorn, meanwhile, dropped his tee ball within 20 feet of the cup.
Howell, who also lost in a playoff here to Thomas Levet in 2001, blasted out of the bunker to 12 feet. Bjorn left his birdie putt short, left of the cup and kicked in his short par putt. Howell then missed his par putt giving Bjorn his eighth victory on the European Tour.
'It's been a long time coming,' said Bjorn, whose last win came at the 2002 BMW International Open. 'I kept perfectly cool over the whole weekend and in the end it paid off. I don't think I've been as relaxed as I was over those two playoff holes.
'I actually try to come out and have a laugh and enjoy myself and try to take the good with the bad. I am just happy playing the game the way I am.'
Overnight leader Michael Campbell was unable to recover from a double-bogey on the 13th. He posted a 1-over 73 to end at 5-under-par 283. Soren Hansen and Simon Khan each shot rounds of 5-under 67 to share fifth place. They were joined at minus-4 by Steve Webster, who carded a 71 on Sunday.
Bjorn trailed the leaders nearly the entire round. He opened with a birdie on the fourth to get to 3 under. However, the Dane tripped to a bogey on the par-4 sixth on the Arden Course at the Marriott Forest of Arden Hotel & Country Club.
The 34-year-old caught fire from there. Bjorn drained a birdie try on the eighth. He made it two straight as his birdie attempt at the ninth found the bottom of the cup.
Bjorn moved to 5 under as he birdied the 10th to make it three in a row. He slid back to minus-4 with a bogey on the 13th. Bjorn jumped into the lead with an eagle on the par-5 17th, a hole he played at minus-5 for the week. He then parred the last to end at 6 under.
Coming to the 18th tee in regulation, the tournament was in Howell's hands. He played the first 14 holes at even-par with one birdie and one bogey. The Englishman birdied the 15th and 16th to get within one stroke of Bjorn.
Howell, like Bjorn, eagled the 17th to jump into the lead at minus-7. However, Howell missed the green right in a sand trap at the last and was unable to save par.
'All the stuff I have been working on with my swing just didn't stand up to the pressure there at the end,' said Howell. 'Simple as that. I got lucky there at the 17th. I made a poor swing with my second shot, but managed to chip it in. I just needed one good swing on the last and just couldn't do it.'
Davis parred the first six holes before an eagle on No. 7 got him within one stroke of Campbell. Davis tied Campbell in the lead at minus-6 with a birdie on the ninth.
However, the Englishman stumbled to back-to-back bogeys from the 10th to fall back to minus-4. Davis recovered those lost strokes with birdies on Nos. 13 and 14. He slid back to minus-5 with a bogey on the 15th.
Davis, playing alongside Howell, birdied the 17th to get to 6 under and got into the playoff thanks to Howell's bogey at the last.
'It was a real struggle today,' said Davis. 'The swing was all over the place. I didn't have the speed on my putter all day and it was a real battle. I hung in there. But I had my chances today and just didn't get the job done.'
Darren Clarke fired a 5-under 67 Sunday that included a double-eagle on the par-5 third. He ended the tournament at 3-under-par 285 and was joined in a tie for eighth by Maarten Lafeber and Robert-Jan Derksen.
Barry Lane, the 2004 champion, finished in a tie for 11th at minus-1 after a closing 68.