Cabrera, who wound up at 5-over 285, won his country's first major golf championship in 40 years.
Woods and Furyk tied for second place at 286.
Looking for his 13th major, Woods closed with a 2-over 72, preserving his record of never having come from behind to win a major. He had a chance to tie Cabrera with a 25-foot birdie putt at the 18th, but his putt curled a foot past the hole.
Furyk shot an even-par 70 Sunday and finished second at the U.S. Open for the second straight year. He is the first player since Arnold Palmer in 1966-67 to do that.
Roberto De Vicenzo won the 1967 British Open at Hoylake for Argentina's only other major championship.
Cabrera had back-to-back bogeys on the 16th and 17th holes Sunday but posted his second 69, in a week during which there were only eight rounds under 70.