The two-time US Open champion followed his record-equalling second-round 63 with a four-under 66 on Saturday, lifting the American to 12 under and two strokes clear of nearest challenger Adam Scott.
Jon Rahm fired a bogey-free 66 to share third spot with Rickie Fowler and overnight leader Gary Woodland, while Woods sits a further stroke back in tied-sixth thanks to a four-under 66.
Koepka began two strokes back but quickly moved ahead by following a tap-in gain at the first with another birdie at the next, as Woodland bogeyed the second after finding water.
Woodland birdied the third but lost his lead when Koepka drained a 20-footer at the fifth, before the pair exchanged gains at the eighth and a two-shot swing at the next saw Koepka make birdie as Woodland made bogey
Koepka reached the turn three ahead but saw his lead increase further when Woodland triple-bogeyed the 10th, only to watch his compatriot birdie the 12th and 14th.
Back-to-back bogeys from the 14th saw Koepka’s lead cut to one, before a two-putt birdie at the 17th doubled his advantage over Scott – who mixed six birdies with a sole blemish during his round.
Rahm birdied two of his final three holes to get to nine under, with Ireland’s Shane Lowry a further shot back after birdieing the last to get within four of the lead.
Woods moved inside the top-10 with five birdies in his first eight holes, as Rory McIlroy slipped further out of contention with an erratic one-over 71.
Matt Wallace fired a hole-in-one at the par-three 16th on his way to a two-under 68, with fellow Englishmen Eddie Pepperell and Ian Poulter also in the group seven off the pace.