Cook, the 11-time PGA Tour winner who has never won a major championship on any tour, completed three rounds at 6-under 207 for a one-shot lead over second-round leader Bruce Vaughan, who had four straight bogeys Saturday to finish the round at 69.
Eduardo Romero (68) and was another stroke back at 4-under 209. Defending champion Tom Watson and Bernhard Langer, who both had 71s, shared fourth place at 212.
No other player was within six shots of the lead.
Vaughan, who had a one-shot lead overnight, appeared well ahead after an eagle at the par-5 fourth and birdies at the sixth and eighth. But the stretch of bogeys between Nos. 9 and 12 allowed Cook to creep into the lead.