Spieth, world number three and twice major champion, putted smoothly to pick up five birdies and did not drop a shot after fierce crosswinds buffeted the early starters.
"That was one of my top five rounds at a major," the 23-year-old told reporters. "I'd give myself a nine (out of 10)."
Koepka, who won the U.S. Open last month, hit three successive birdies from the 11th hole before making his only bogey at the 16th. He chipped in from a pot bunker to eagle the par-five 17th and return to the top of the leaderboard.
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England's Ian Poulter, runner-up in the 2008 Open on this course, managed four birdies and one bogey in a solid round of 67.
The Ryder Cup specialist, who has never won a major, endured a frustrating 2016 due to injury problems and had to come through qualifying to earn his place in the tournament.
"I've definitely had some low spots in the last 18 months and I was getting very down," the 41-year-old told reporters.
"I'm proud of the way I've been able to get things back on the straight and get back to really focussing hard on what I need to do to get the level of golf back that I think I can play."American Justin Thomas, sporting a black tie, matched Poulter's 67 to finish a shot ahead of world number two Hideki Matsuyama of Japan, Sweden's Alex Noren and world number 520 Stuart Manley of Wales.
Defending champion Henrik Stenson of Sweden finished on 69, along with England's Matthew Fitzpatrick and Belgian Thomas Pieters.
Many players struggled in the conditions, however, and American Mark O'Meara endured a day to forget on his return to the course where he won the 1998 Open.
Handed the honour of hitting the first shot of the tournament, O'Meara sent his drive out of bounds before running up a quadruple-bogey eight and he ended up signing for an ugly round of 81.
World number one Dustin Johnson, four-times major champion Rory McIlroy and American Phil Mickelson, runner-up to Stenson last year, were among the late starters with conditions expected to ease.
(Editing by Christian Radnedge)
