After two underwhelming even-par rounds, Woods needed something special in the third round at Carnoustie and he was on course for just that after reaching the turn three-under for the day.
Better was to come with an incredible approach to the 10th setting up another birdie, followed by another at the 11th, leaving Woods five-under, just one stroke behind joint-leaders Zach Johnson and Kevin Kisner before they started their rounds at 4pm local time (1500 GMT).
Reigning champion Jordan Spieth also moved to five under par after an eagle at the par-four first hole which signalled his intent, although fellow American Kevin Chappell climbed into the joint lead on six-under after birdies at each of the opening three holes.
The later starters were hoping the afternoon conditions would remain as calm as the morning, when Rose fired a superb seven-under-par round of 64 to move to four-under overall.
Rose's effort equalled the lowest round in an Open on the Scottish links, although it was not quite good enough to trouble Tommy Fleetwood's overall course record of 63, set last year.