Craig Stadler, the second-round leader, posted a 1-under 70 to take second place at 2-under-par 211. Two-time British Open champion Greg Norman (70) shares third place with Irishman Des Smyth (68) at even-par 213.
Watson, a five-time British Open champion, birdied the par-3 third to open his round. He came back with a birdie on the par-4 fifth at Royal Aberdeen Golf Club to get to plus-2.
The 55-year-old then eagled the par-5 sixth. Watson moved into red figures with a birdie on No. 7. He tripped to a three-putt bogey on the seventh, but that was his only mistake on the day.
Watson, who owns 11 major titles with eight on the PGA Tour and three on the Champions Tour, parred his next three holes before tying Stadler for the lead. Watson birdied the par-5 12th for the second straight day to join Stadler in first at minus-1.
The six-time winner on the Champions Tour moved ahead of Stadler with a birdie on the 14th. Watson got to 3 under when his birdie putt on the 16th found the bottom of the cup. He ended his round with a pair of pars for a 64, that matched Cooper for the low round of the championship.
'Today was the first time in a long time I made some putts,' Watson said. 'I hit the ball well today, the best I've driven it in two days. I may play a little bit differently tomorrow, as far as where I try to place the ball off the tee.'
Stadler stretched his early lead to three shots with a birdie on the par-5 second. He moved to minus-3 with a birdie on the eighth.
The 52-year-old fell back to minus-1 after a bogey on nine and another one at 12. Stadler got one stroke back with a birdie on the 14th, but stumbled to the third bogey of his round at the 15th.
Stadler climbed back within one stroke of the lead with a birdie on the par-3 17th.
Mark James, the 2004 Ford Senior Players champion, shot a 1-under 70. He shares seventh place at 6-over-par 219 with Ian Mosey, John Bland and Seiji Ebihara.
Mark McNulty, who owns four Champions Tour titles, is one shot further back at plus-7. He was joined there by Horacio Carbonetti.