Stenson moved ahead of the pack with a birdie on the par-4 14th and clinched his fifth European Tour title with a birdie on the par-5 18th.
The Swede closed with a 4-under 68 to finish the tournament at 19-under-par 269. Third-round leader Ernie Els, a three-time winner here, managed a 1-under 71 to end alone in second at minus-18.
Ross Fisher, who held at least a share of the lead after the first two rounds, closed with a 71 to finish in fifth at minus-16.
Fisher joined Woods, Els, Fasth and Stenson as the five players who owned at least a share of the lead during the final round. However, Fisher was done in by back-to-back bogeys from the 16th.
Stenson, who now lives in Dubai, birdied the par-4 second for the fourth straight round and that gave him a share of the lead with Els, who bogeyed the same hole.
The 30-year-old Stenson remained at 16 under as he parred the next six holes. During that span, Els and Fisher pulled ahead before eventually falling behind.
After the last of his six straight pars, Stenson was one up on Woods, Els and Fisher. However, Stenson faltered to a bogey on nine to join those three atop the leaderboard at 15 under par.
Stenson birdied the 11th for the third day in a row, joining Fasth, Fisher and Els in the lead. Fisher, playing one group ahead of Stenson, birdied the 13th. Stenson also birdied 13, for the fourth straight day, to tie Fisher at 17 under.
From there, Stenson pulled one clear with a birdie on No. 14 and things stayed that way as he parred the next three holes. Stenson drained a 7-foot birdie putt on the par-5 18th to best Els, who also birdied the last.
'It's unbelievable to win sort of your own tournament,' Stenson said. 'I came so close in Sweden two years ago when I lost in a playoff for the Scandinavian Masters. This is my second home, so to be able to win in front of family and friends is just unbelievable.
'It's always nice to beat Tiger, obviously. It's the first tournament I have won that he's been in. He is a great player, so of course that gives you extra satisfaction to beat him. To play four rounds with Ernie and to beat him by a shot coming down the stretch is the same satisfaction.'
Els led by three entering the round, but tumbled down the leaderboard with three bogeys in the first six holes. Birdies on seven and 10 got Els back into a tie for the top spot at 16 under with Fisher.
The South African moved to even-par for his round with a birdie on 14. Els birdied the last, but it was not enough to force a playoff.
'I knew after the start, I just wanted to give myself a chance coming down the stretch,' admitted Els. 'I really got out of sorts on the greens and had a couple of bad swings. I got myself under control after the start and then it was a real battle of who could make putts and who could hit the right shot in there. Henrik was great. One shot separated us for four days.'
Woods, who won his seventh straight PGA TOUR event last week, was 1-over through 11 holes with two birdies and three bogeys. He shared the lead after a birdie on the eighth, but bogeyed 10 and 11 to drop three back.
The 31-year-old Woods rallied on the closing nine with three consecutive birdies from the 13th to get within two at 16 under. However, Woods only managed a birdie at the last for the fourth straight day to share third.
'I hit it well enough to at least give myself a chance, but just did not putt well at all,' Woods said. 'I putted poorly all week, and it's frustrating because I come over here to the best greens we've seen in a long time and I miss a bunch.'
Fasth opened with birdies on one and two. He dropped a shot on the fourth, but climbed into the mix with a birdie on 10. Fasth moved into a tie for the lead with a birdie on 12, but managed just one more birdie the rest of the way.
Peter Hanson shot 1-under 71 Sunday to take sixth at 14-under-par 274. Simon Dyson, Prom Meesawat and Jyoti Randhawa were one stroke further back at minus-13.
Thomas Bjorn, the 2001 champion, shared 10th place at 12-under-par 276 with Robert Karlsson, Maarten Lafeber and two-time runner-up Miguel Angel Jimenez.