Molinari managed to produce a two-under final round of 69 on the Scottish green to finish eight under, ahead of four players, including locals Rose and McIlroy, who tied on six under. Woods finished five under after excelling midway through the round, with Jordan Spieth on four under.
Molinari, 35, overtook a six-way tie for the lead, to collect the Claret Jug. He had started his final day three shots away from leaders Spieth, Kevin Kisner, and Xander Schauffele, but held his own while the others missed shots in the dispute for the lead.
"It is absolutely amazing. I think it will take a long time to sink in. It has been a great week,” Molinari told the BBC. "The course bit me a few times in the first two days, but to go bogey-free around this track at the weekend is incredible."
Molinari, who is ranked fifteenth in the world, had impressed experts before The Open despite his reserved Open record. In the past few months, though he had won the PGA Championship at Wentworth and the PGA Tour at the Quicken Loans Classic. He began the weekend with opening rounds of 70 and 72 before an impressive 65 on Saturday.
On Sunday, his streak continued with 13 pars in a row. The sequence ended with a birdie on the par-five 14th, before a stunning approach on the 18th garnered him another shot. Molinari’s closest challenger, Schauffele, 24, had to eagle the last hole to force a playoff, but he landed short, awarding Molinari the win.
Woods, who won his first major in 1997 at the age of 21, had two birdies on the front nine on Sunday. These along with dropped shots by Spieth and Schauffele enabled him to take the lead at a major for the first time in years. However, a double bogey at the 11th ended his chances of winning the Claret Jug. He ended up sharing sixth place after signing for a level-par 71.
The UK’s Rose and McIlroy had the home green advantage and were in good standing before Sunday, but their chances were derailed after Molinari took advantage of the 14th to take the lead. Meanwhile, Spieth had hoped to defend his title, but after both he and Schauffele dropped three shots between the fifth and seventh holes, their chances at a championship were extinguished.
"We were in the strangest spots possible on the golf course, where we didn't think we would be," said Schauffele. "Every time I looked up at the leaderboard, there's four, five, six guys in the lead, and five, six guys one back. I figured looking at the board isn't going to do me any good and I was just happy to claw my way back in a little bit."