Johnson finished a stroke clear of veteran compatriot Mark O'Meara and Spain's Rafael Cabrera-Bello with Miguel Angel Jimenez, Dustin Johnson, Brandt Snedeker, Tom Lehman and Shiv Kapur a further shot back.
World number one Tiger Woods, among the early starters on Friday, scrambled to a two-under 69 in the first round, leaving him well placed to challenge for his 15th major title over the weekend.
Among those players struggling to make the cut will be twice major champion Rory McIlroy, who slumped to an opening 79, and former world number one Luke Donald who started with an 80.
The weather is expected to stay warm and tournament organisers must decide whether to water the parched greens or make the pin positions more accessible following complaints from a number of players on Thursday.
Spain's Sergio Garcia urged the Royal & Ancient to keep the putting surfaces the same as they were in the first round.
"I'd say they were the fastest and firmest links greens I've played," he said after carding a 75.
"What will be unfair is for the guys who play in the afternoon on Friday to play them soft when they've played them like this."
(Writing by Ed Osmond; editing by Tony Jimenez)
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