Overnight leader Hale Irwin faltered with a 1-over 73 and dropped to three off the pace, where hes joined by Fuzzy Zoeller (69).
Morris Hatalsky (67), Allen Doyle (70), Larry Nelson (72) and Bob Gilder (73) are all at 5-under-par, four back of the leaders.
Watson, who won the Senior Tour Championship in 2000, has had five second-place finishes this year and hopes his position going into Sunday will be enough to get him over the winless hump.
After starting with a birdie on the first, Watson used an eagle-2 at the par-4 seventh to vault him into the lead.
'It drew back a little bit, hit the flag but stayed up,' said Watson, who needed just 23 putts during his bogey-free round. 'Even with my bad eyes, I still thought I saw the ball on the green there. That was the highlight of the round.'
He then followed that up with another birdie at the eighth to go to 7-under.
Meanwhile, Irwin, who leads the Senior Tour in birdie average, went an amazing 24 holes without one before getting a birdie on the 17th. He quickly gave it back with his second bogey of the day, on a par-5 18th. He also bogeyed the par-5 third.
Morgan, the 1997 champion, began the day just a shot behind Irwin and finally got his round going with a strong start on the inward nine. Birdies at the 10th and 11th and another added at the 13th found him tied with Watson at the top before settling for pars at the last five holes.
'It seemed like I didn't play quite as well as I played the first two rounds, especially on the front side,' Morgan said. 'I had some opportunities and let them get away. That's part of the game.'
Zoeller was 3-under through eight to get into the mix, but couldnt muster another birdie the rest of the way. It was his second-straight 69.
The round of the day belonged to Hatalsky, who broke through earlier in the year with a win at the United Fore Care Classic. Three consecutive birdies starting at the sixth moved him up the leaderboard. Three more birdies and two bogeys left him in a tie for fifth.
Gilder, who played alongside the equally disappointed Irwin, had two bogeys and a lone birdie.
Full-field scores from the Senior Tour Championship