But Faldo faded, finishing with a 73 for a four-round total of five over par.
He finished with a four-round total of seven-under 281.
Woods' four-round total of seven-under 277 had him stuck firmly in the middle of the pack.
His four-round total of 282 broke a 20-year-old Open record.
Toms finished with a four-round total of 19-under-par 261 and earned a $918,000 first-place check.
A third-round 65 put him atop the leaderboard and he coasted to victory with a final-round 69 for a 20-under 268 four-round total.
Gary Player won at Carnoustie in 1968 with a four-round total of 291, the highest in the last 50 years.
Turner had a four-round total of 15-under 273 and earned $48,750.
That gave him a four-round total of 267, 13-under par.
He finished with a four-round total of 16-under 268, with Russ Cochran the runner-up.