He joined Roberto di Vicenzo, winner of the 1967 British Open at Hoylake, as the only Argentinians to win one of golf's four major titles.
Furyk posted an even-par 70, while Woods, failing to display the clinical precision that saw him march up the leaderboard on Saturday, settled for a two-over 72 to join Furyk on 286.
Sweden's Niclas Fasth (70) was fourth on 287.
"Angel played a beautiful round of golf," Woods said. "He put pressure on Jim and I, and we just could not get it done."
Angel Cabrera joined Roberto di Vicenzo as the
only Argentinians to win a major [Reuters]
The 37-year-old Cabrera had claimed a three-shot lead with a birdie at 15, where he almost holed his second shot.
But back-to-back bogeys at 16 and 17 opened the door for Furyk and Woods.
After bogeys at 11 and 12, Furyk had worked his way up the leaderboard with three birdies in a row from the 13th.
That run saw Furyk leapfrog past world No.1 Woods.
But Furyk bogeyed 17 to fall out of a tie for the lead, and a poor approach ended any chance to force a playoff.
Woods, meanwhile, was battling for his life as Cabrera watched on television.
"I was just hoping they would not make any birdies," said Cabrera, the world No.41 who improved on his previous best finish in a major - a tie for fourth at the 1999 British Open.
"Tiger can birdie any hole."
From a bogey at 11, Woods posted seven straight pars, but he just couldn't get the birdie he needed.
An attempt to drive the green of the 306-yard 17th saw him find a bunker, from which he settled for par.
At 18 his tee shot was nestled in short rough to the right and his second shot left him more than 20 feet from the hole, ensuring that a rare blot on his resume remained intact - despite 12 major triumphs Woods has never won one of golf's grand slam tournaments in which he trailed heading into the final round.