The three-time European Tour winner, who only turned 50 two months ago, birdied the second before eagling the 580 yard seventh at Colorado Golf Club.
He birdied the 11th, but his round threatened to unravel with a double bogey six at the next.
However, four more birdies and just a solitary dropped shot on over the closing holes took the Scot to one under for the week, five behind leaders Tom Lehmann and Jay Don Blake.
Blake produced a two under par 70 to join former Ryder Cup Captain Lehmann (71) on six under.
Fred Couples, Mark O'Meara and Mike Goodes complete an all-American top five in a tie for third two shots off the pace.
South African David Frost and Zimbabwe's Nick Price both occupy a share of ninth on two under, with Eduardo Romero and Bernhard Langer still in contention at level par.
Frost, who was playing on The European Tour only 12 months ago before taking up his career amongst the veterans, shot a course-record 65 and was delighted to give himself a chance going into the final round.
"These things happen in Majors where somebody comes out of the pack early in the day and gets in contention," said Frost.
"More than likely guys don't shoot two low rounds on the weekend after making a big comeback on Saturday, but who knows? You just go out there and play like you have nothing to lose, just play my game.
"As I say, I haven't played as well as I would like to, so maybe I turned the corner, maybe I found something today that I might look for next week."