Woods was chasing his 15th major title - his first since the 2008 US Open - and had the lead halfway through the final round before faltering and finishing three strokes behind the victor, Francesco Molinari.
One of the other contenders was 14-time major victor Tiger Woods, who was back vying for one of golf's biggest prizes again before finishing three shots behind Molinari.
"I didn't get off to a great start but I hung in there and battled back and just ran out of holes at the end", he said.
"That was too close to the game of play", he said.
'It's amusing when you look back, how Tiger has been involved in so numerous greatest moments for Italian golf, ' he said. "It was nice to be a part of it".
I can not believe there are actually people who think Tiger Woods should not be picked for the Ryder Cup.
Molinari, 35, underlined his status as the hottest player in world golf by finishing with an eight-under total of 276 helped by a fantastic 65 on Saturday.
Tiger Woods has joined Kevin Chappell and Francesco Molinari for company on -6, after making a fine birdie on the fourth hole. His appearance on the 18th Hole made news when a spectator yelled out just before Woods hit the ball.
Number of bogeys Francesco Molinari made in his final 37 holes, almost unthinkable given the pitfalls that await during every trip 'round Carnoustie, among the hardest links courses in the world.
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"I knew I was coming in playing some good golf but my record round here was bad".
Francesco Molinari's sensational Open Championship win has catapulted the Italian to the top of the European Team Ryder Cup standings and he's now a lock to be facing the Americans on September 28 in Paris.
"I'm relearning. I feel like I've won quite a few more recently than he has, but you still have to relearn to deal with it".
For much of his career, if Tiger Woods said he had a shot to win a major golf tournament, you believed him. I have no regrets.
With that triumph came extra fame, media attention, prestige and winnings - he took back to Italy $1.8 million after topping the Carnoustie leaderboard.
'He knows how hard I've been working to get here so I think he felt the same proudness I felt yesterday.' Molinari revealed the Claret Jug had spent the night on his bedside table, although he did not get much sleep as he celebrated his first major title. I played with him the final day at Wentworth and he didn't miss a shot.
"This was his first taste of major championship drama for quite a while", McIlroy said.
"So I can look at it - anyone can look at it however they want to, but I'm going to look at it as a positive moving forward, and try to learn how to handle the situations a little better next time".