Fasth held a two-shot lead through 16 holes, but closed his final round with consecutive bogeys to finish with a 3-under 69. That put him at 18-under-par 270, which is where Bickerton was in the clubhouse after a course record-tying 63.
Both players parred the first three playoff holes -- each played at the 18th -- before Fasth rolled in a 5-foot birdie putt at the ninth hole to wrap up his fourth European Tour title.
'It was a bit more work than I was hoping for,' said Fasth, who also won playoffs for his two victories in 2005 (New Zealand Open; Deutsche Bank Players Championship). 'I really had it in my own hands and messed it up a bit.'
Fasth, 34, had six birdies and three bogeys on Sunday. Bickerton's 63, which tied the San Roque Club course record set by Robert Karlsson on Friday, was bogey-free.
'It's been quite strange really because I have put a new driver and new irons in my bag this week, and my old putter, and it's all come to fruition,' said Bickerton, who is winless in two career playoffs.
Thomas Bjorn held the lead for a spell on Sunday, but finished with a 3-under 69 to end in six-way tie for third place at 17-under-par 271. Bjorn missed a 5-foot par putt at the 18th that would have also tied him with Fasth and Bickerton.
Overnight leader David Griffiths carded three bogeys on the front nine and three birdies on the back, ending with an even-par 72 to also finish at minus-17.
Joining Bjorn and Griffiths in third place were Philip Archer (67), Mattias Eliasson (67), Jose Manuel Lara (68) and Gary Orr (67).
Fasth originally snatched the lead from Bjorn with back-to-back birdies from the seventh, moving to 18-under with a 6-foot putt at the eighth.
Playing six groups ahead of Fasth, Bickerton chipped within a foot at the par-5 16th and made birdie to get within one shot, then used a nice tee shot to set up another birdie at the par-3 17th.
After that, Bickerton, who began the day eight shots off Griffiths' overnight lead, was tied with Fasth at minus-18.
Fasth got as low as 20 under in his round after making birdies at 12 and 16, but his round began to unravel at the 17th, where he pushed a short par putt right after missing the green on his approach shot.
At the 18th, Fasth drove into the trees next to the fairway, then knocked his approach shot into the gallery on the way to another bogey. The par putt he missed was from about eight feet.
Explained Fasth, 'I missed the green just on 17 with a 5-iron. I got a bad bounce, but it happens. I missed the tee shot on the last and didn't get up and down.
'They were straight forward bogeys,' Fasth added, 'but not where you want them.'
Back at the 18th for the first playoff hole, both players made par -- Bickerton after a pin-high approach shot that landed within 20 feet, and Fasth on a two-putt from about 35 feet.
It was the same story the second time around: Bickerton found the front of the green and two-putted for par, while Fasth landed in the middle and also two- putted.
Both players then missed the green on the third playoff hole, but made par anyway before heading to the par-4 ninth.
Bickerton's approach shot at the ninth just missed water, landing left of the green and in the hospitality area. Fasth, meanwhile, knocked a 9-iron over the water and within five feet of the hole to set up birdie.
After a free drop, Bickerton played his chip shot to within 4 feet, then stepped aside as Fasth holed his birdie putt for the win. Fasth, who deliberated for several minutes before he attempted the putt, pumped his fist when it found the center of the cup.
'I have bounced back many times this week,' said Fasth. 'I have had my setbacks, but I have come back. I didn't really feel great about my shots in the playoff, but I hung in there and eventually hit a great shot and made birdie.'
Carl Suneson finished alone in ninth place at 16-under-par 272 after a final- round 69. Karlsson shot a 72 and was 10th at minus-15, one shot ahead of 2005 champion Peter Hanson, Juan Parron and Miles Tunnicliff.