The 50-year-old American had all pars on the front nine for a 34 at sunny Royal Birkdale. He then birdied the 10th and bogeyed the 11th. His three successive birdies from the 15th left him at 33 for the back nine.
''I hit a lot of solid shots in the wind, so it didn't really mess with me too much,'' Sauers said. ''I'm really pleased with the way I played. This is only my third British Open and it's a great privilege and honor to be here.''
Four players were at 68, among them 2010 British Senior Champion Bernhard Langer. The German was joined by South Africa's David Frost, Australia's Peter Senior and Frankie Minoza of the Philippines. Only 10 of the 144 players bettered par.
Sauers is 23rd on the Champions Tour money list and has made 11 cuts from 11 starts this year with two second-place finishes and a third.
Defending champion Fred Couples and Mark O'Meara, who won the 1998 British Open at Birkdale, shot 74s. Colin Montgomerie, their playing partner and a senior rookie, was two strokes better.
Langer birdied three of his last four holes.
''I was very pleased with my long game and finally the putter started working at the end,'' he said.
Frost picked up a birdie at the sixth to be out in 33. He birdied No. 10 to go 2 under but then dropped shots at the 13th and 16th before an eagle at the long 17th got him back in 35.
Senior moved to 3 under with birdies at Nos. 10, 11 and 17 but dropped a stroke at the last hole.
''I think there will be a lot of good scores out there,'' Senior said. ''The course is playing nice and it gives you a lot of opportunity.''
Minoza fought back from 4 over after just three holes to post the early clubhouse lead at 68. He had birdies on Nos. 10, 14, 15, 16 and 18 to finish the back nine in 31.
Cook was undone by a double-bogey at the last hole, turning a potential 67 into a 69.
''It was one of those terrible mistakes that you make and learn from. But I'm pleased to be under par in a major championship,'' he said.
Hallberg, second to Fred Couples at Turnberry last year, had an eagle, four birdies, three bogeys and a double-bogey on his card.
''That's my game,'' he said. ''It's up and down. But it was just a joy to play out there. It was beautiful.''