His playing partner, Francesco Molinari keep his head down and continued to make pars on Sunday.
That vaulted McIlroy into a six-way tie for the lead and Rose joined the expanding group on six under with a tap-in birdie on 18, two days after he needed a birdie on the same hole just to make the cut.
When Woods proclaimed that he had a shot to win The Open Championship after being within striking distance of the lead, it was a little bit tougher to buy in.
Tommy Fleetwood also started his round at 5-under, like Woods, and birdied the first to a massive ovation.
"It's pretty emotional because they gave me some pretty significant hugs there and squeezed", Woods said.
"To me, it's just so special to have them aware, because I've won a lot of golf tournaments in my career, but they don't remember any of them".
With defending champion Jordan Spieth, Woods, English duo Rose and Tommy Fleetwood and McIlroy in the Sunday mix, the Molinari sub-plot was nearly overlooked as the leading groups went out.
"The beginning of the year, if they'd have said, 'You're playing The Open Championship, ' I would have said I'd be very lucky to do that".More news: Los Angeles shootout: One dead, suspect held after taking hostages at supermarket
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Molinari twice saved par with 8-foot putts on the 12th and 13th, and he took the lead for the first time with a short birdie on the par-5 14th. I hit really good putts, too.
Woods sent his second shot into the gallery, with the ball rebounding out into the rough. The real test comes Sunday, when he'll have to find a way to come from behind and actually win the Open - something he didn't do even in his prime.
Kevin Chappell made two double bogeys, the last one on No. 17 that derailed his hopes.
It was Molinari who didn't make a mistake, shooting a final-round 69 and playing the final 37 holes of the tournament without a bogey.
The Italian attacked the course well on Sunday as he tried to capitalise on the errors by the three overnight leaders; Jordan Spieth, Xander Schauffele and Kevin Kisner who all dropped shots. Schauffele chopped up the next hole for double bogey.
Both players appeared to have lost their chances of threatening the top of the leaderboard until they each made an eagle at the par-five 14th - Carnoustie's easiest hole - to move back into the reckoning. It is Woods's best finish in a major since the 2013 Masters, and certainly another positive step in this comeback from the deepest depths.
Englishman Eddie Pepperell says he was a "little hung over" during his round of 4-under 67 that took him to within one shot of the lead at the British Open. But a T-2 at the PGA Championship a month later, plus his career year in 2018, have validated a lot of hard work.
"I feel fine", Spieth said.
'The only thing they've seen is the pain I was in and the struggles I went through. "So I could use it to my advantage, and, just did a great job at that".