The American left-hander, who holed out from 30 yards to eagle the par-five eighth, birdied four of his last seven holes to hold off a late surge by Britain's Chris Wood at Dove Mountain.
In a match spread over two days after driving sleet and heavy snow forced play to be suspended on Wednesday, Watson sank an 18-footer at the 16th and then struck his approach to within three feet at the 17th to seal victory.
"Very satisfying," a beaming Watson said after advancing to the second round where he will meet fellow American Jim Furyk, a 4&2 winner against Ryan Moore.
"I missed the cut last week, so I just worked on my game, worked on my mental focus and hit some key shots down the stretch. We both played better this afternoon.
"Yesterday with the cold weather we all struggled a little bit. We both made birdies and we just kept trying. Luckily I made some birdies down the stretch," added Watson, whose missed cut came at last week's Northern Trust Open.
Englishman Wood, who won his maiden European Tour title at the Qatar Masters in January, piled up five birdies in his last 10 holes but paid the price for a poor start in difficult weather conditions the previous day.
"It was crazy," Watson said of the rain, sleet and snow which led to play being abandoned on Wednesday. "It showed in our match yesterday. We didn't make a birdie.
"We gave some holes here and there because of the cold weather, your hands, your feel, everything. And then today I made an eagle and four birdies, and he made five birdies. Today was real good golf."
Asked to assess the task he faced in the second round against 2003 US Open champion Furyk, Watson replied: "Jim Furyk is one of the greats of the game.
"He's a great competitor and he's going to think his way around the golf course. And I'm not a great of the game.
"I'm probably just going to beat it and slash it around and hopefully make some birdies and somehow come out on top," said Watson, who is one of the biggest drawcards in the game with his prodigious length off the tee and his creative shot-making.