Woods was six off the lead at the halfway stage after consecutive rounds of 71, and he made a solid start to the third round as he opened with three pars before beginning his march up the leaderboard with a birdie at the fourth.
He finally managed to birdie the long sixth for the first time this week, and the roars of the huge gallery following the three-time champion got progressively louder as he reeled off three straight birdies around the turn to pull within one of the lead.
The 42-year-old then found himself in a tie at the top when he hit the 14th green in two and safely two-putted for his sixth birdie of the round, but he finally faltered at the 16th when his tee shot failed to reach the back tier and he needed three putts to get down.
Woods appeared likely to blot his card again at the last, where he pulled a long iron from the tee too far left and watched anxiously as his ball held up in the thick rough just a couple of feet away from the Barry Burn.
But, after gouging his ball back into play, Woods clipped a delightful third to three feet and rattled in the par-saving putt to stay at five under heading into the final round.
His 66 represented his best round in the tournament since his last lifted the Claret Jug at Hoylake in 2006, and he was confident about his chances of contending for his 15th major title on Sunday.
“That was good, I played well today,” said a delighted Woods afterwards. “I hit a lot of good shots, and I really didn’t feel like I made a bad swing until 18. I really felt like I had control of the golf ball today and, on top of that, I made some longer putts, which was nice.
“There were a bunch of guys that were putting up great scores, and the golf course was gettable. I didn’t want to be too far back if the guys got to 10 under par today. I had to stay within reach. And five under is definitely within reach.
“It would be nice if there weren’t a lot of guys between myself and the lead, but not the way this golf course was going to set up. We knew there were going to be 10 or 12 guys with a chance to win on Sunday, and it’s turning out to be that.”
Woods, who is playing in The Open for the first time since 2015, also stressed the importance of his par-save at the final hole having come so close to finding the hazard with his errant tee shot.
“That was big for me just to not finish with two bogeys on the last three holes, playing as well as I did,” he added. “As I said, I really didn’t hit a bad shot until 18.
“I had to lay up on the left side so I had an angle at that flag, and I only had 83 yards. I figured just this nice little one like I practised in the backyard, and I hit my number.”