Vaughan, who has undergone three partial knee replacements over the past two years, totaled 3-under 139. John Cook (71) is one stroke back.
"It is my left knee which is why, when it hurts a lot, I don't tend to play so well," said Vaughan, the overnight co-leader. "But when I went out today, I looked at the scores and I thought that level par would be a pretty good number. So I am pretty pleased with that."
The Kansas native had two bogeys, which he followed with birdies. The former firefighter, who took up golf at 20, is aiming for first major win on the senior tour.
Tom Watson (71), Bernhard Langer (71) and Eduardo Romero (73) were another stroke back under windy conditions at Royal Troon links. Greg Norman shot a 72, including an eagle at the 16th, to finish eight strokes off the leader.
On Sunday, Norman led the British Open with nine holes to go before finishing tied for third behind Padraig Harrington.
Watson made his first three bogeys of the championship. But Watson said he was lucky to walk away with a bogey the par-4 11th after his tee shot landed in a bush for an unplayable lie.
"I pitched just short of the green and then pitched over the bunker to 12 feet and made that for a bogey 5," he said. "I thought I would have made 8, but I went from 8 to 5 in one hole."
Langer, a two-time winner in his first season on the senior tour, made a 30-foot birdie putt at the ninth and pitched within 2 feet for another birdie at the 15th.
Langer said he's playing as well as he did on the PGA Tour.
