The 18-year-old from Oslo fired a one-under-par 71 at sunny Hoylake to finish on even-par 288, seven strokes ahead of Italy's Edoardo Molinari.
Molinari, the only other amateur who made the halfway cut, closed with a 75.
Thorp holed a 25-footer to birdie the last and finish level with playing partner and five-times Open champion Tom Watson.
"This is fantastic," said the Norwegian, who booked his place at Hoylake by winning last year's European amateur championship.
"I played really well all four days and today I played with one of the greatest legends in the game. It's been a perfect day.
"Now I'm going to take a few weeks off now, having played two months in a row. I'm going to play the European Tour qualifying school later in the year and get some experience."
Watson was impressed by Thorp.
FEARLESS PUTTER
"I think he's a very good player and he's fearless with the putter," the 56-year-old American said. "I remember those days. God I wish I could do that now."
While Thorp birdied three of the last eight holes in breezy conditions, world number one and champion Woods prepared for a 1330 GMT start in the company of Spaniard Sergio Garcia.
The 30-year-old American holds a one-stroke lead going into the final round of the year's third major after grinding out a 71 on Saturday.
Chasing an 11th major title, Woods remains the red-hot favourite at 13 under par but has four players in the world's top 10 hot on his heels.
Ernie Els, the 2002 champion, is tied for second place with Garcia and American Chris DiMarco, Jim Furyk is a further stroke back in a share of fifth with Australia's Adam Scott, the world number six, at nine under.
Britain's Paul Broadhurst, boosted by eagles at the fifth and 10th, was the leader in the clubhouse at four-under 284 after firing a 69.
The 40-year-old Englishman raced to the turn in 32, chipped in from off the green on 10 and partially recovered from double-bogeys on 11 and 14 with a birdie-par-birdie finish.
