The 37-year-old Englishman needed to hole a nerve-shredding 14-foot birdie putt in Friday's second round to avoid missing the halfway cut but, galvanised by his narrow escape, he ended on a high with closing rounds of 64 and 69.
"It just proves to me I can play well in this tournament, that I can win the Open," world number three Rose told reporters after a six-under aggregate of 278 gave him a share of second place, two strokes behind winner Francesco Molinari of Italy.
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"When I'm in the hunt I enjoy it. I play my best golf. I don't back away."
Rose thanked the Scottish galleries for helping him to surge through the pack over the weekend.
"It was great to get the crowd behind me," he said. "I hadn't felt the energy of the crowd for a while in the Open.
"That was a real positive for me and it renews the love of the Open for me."
Rose, still waiting to add to the one major victory he has claimed at the 2013 U.S. Open, ignited his challenge for the Claret Jug when he eagled the par-five 14th.
A radar-like approach shot to the 18th then left him a six-footer for a birdie and, after the ball nestled in the cup, he turned and hailed the applause of the huge grandstand.
"After birdying that hole all four days, it has to be right up there for me now," said Rose.
"Today it wasn't really playing that bear of a hole that it can. Yesterday I hit driver, 5 iron, which is more the Carnoustie 18th hole that we know. I feel like I birdied it all ways, which makes it special."
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