The 59-year-old Watson, who almost became the oldest major champion at the British Open last weekend, is three shots behind leader Fred Funk.
Funk birdied five of his last eight holes for 64 -- the lowest opening round in the history of the championship.
"The first nine could have been three or four over, and the back could have been eight under," said Funk, who had a two-shot lead over fellow Americans Jay Haas and Loren Roberts, Des Smyth, of Ireland, and Christopher Williams, of South Africa.
Watson was another stroke back and tied with Tom Kite, Jerry Bruner, Greg Norman, of Australia, and Sam Torrance, of Scotland.
Watson said he hadn't watched a replay of his missed putt on the final hole at Turnberry, an 8-footer for par that would have won him the claret jug.
"I don't want to. That was the ugliest stroke in the world. I don't want to relive that thing," Watson said, before admitting he initially wanted to play it over again. "Then I said, 'To heck with it. That's history. That's over and done with.'"
Asked whether he had ever been in so much demand before, Watson replied: "Frankly, no. Just the outpouring of people communicating with me -- I've never had that happen before."
Watson didn't look like any disappointment from the British Open carried over, starting 3-under after five holes and making two more birdies and two bogeys the rest of the way.
"I had birdie chances on the last four holes but didn't make them. It was a good score. Maybe it could have been better."
A heavy shower at the sixth changed the nature of the greens, and Watson struggled with his putting the rest of the round. He missed an 8-footer at the 13th -- the same distance he missed on the 72nd hole at Turnberry.
"I'm not very good on 8-footers right now," Watson joked.