Northern Ireland's McDowell was a two shot leader at the start of the day and was four shots ahead after two holes before Johnson, twice a winner at Pebble Beach on the US PGA Tour, made his local knowledge count.
The American's five under par 66 gave him the 54-hole lead at six under, three shots better than McDowell, who posted a level par 71 and was not overly disappointed.
"I'd rather be three shots in front, of course," McDowell said. "But I shot 71 not being in control of my game and I've got to be pretty happy with that.
"It's going to be a fun day tomorrow."
Four behind Johnson, Woods showed a glimpse of a return to form as he posted a 66 on the course where he won by a Majors' record 15 strokes at the 2000 US Open Championship and Gregory Havret of France booked a tee time with the World Number One in Sunday's penultimate group after a 69 got him to level par.
Two-time winner Ernie Els is also level after a 72 with Masters Tournament champion Phil Mickelson a shot behind the South African after a rollercoaster 73, two shots ahead of Tim Clark, Alex Cejka and Ryo Ishikawa.
Mickelson had wowed Pebble Beach with his second round 66 and on Saturday it was Woods' turn.
Starting seven strokes behind McDowell, he had been searching for lost form following an on-course struggle of late.
After carding three birdies and three bogeys on his front nine, Woods' struggles seemed to be continuing but he changed gears, delivering five birdies on the supposedly more difficult back nine to come home in 31, closing birdie, birdie, birdie.
Woods had looked back to his imperious best as he negotiated the difficult par three 17th with a big-breaking birdie putt to get to level par and then produced a brilliant second shot from the 18th fairway to inside 15 feet, from where he two-putted.
Woods, though, was not getting carried away with his performance, despite the animated manner in which he had celebrated his closing birdies with caddie Steve Williams.
"I was just trying to get back to even par or one over for the Championship," Woods said. "I thought that would be a nice place to end up and I just happened to do one better than that.
"I just kept telling myself it's a process, you've got to stay patient.
"It's a long haul, the US Open, just keep plodding along. I made a couple of key saves for par, one at nine and you've just got to hang in there and I did that."
McDowell made a flying start with two birdies but bogeyed the third, and when playing partner Johnson eagled the fourth his lead had been cut to one.
Johnson birdied the sixth and then showed he was more than just a big hitter as he sent his wedge tee shot at the 99 yard par three seventh to within a foot for another birdie and the outright lead.
The lead changed hands again at the ninth as McDowell birdied and his rival bogeyed before the American birdied the 11th.
McDowell made a nerveless par save at the 12th but the closing holes bit back.
Johnson bogeyed the 13th and McDowell the 16th before a crucial two shot swing at the 17th saw another bogey for the Northern Irishman as the American birdied.
Johnson recovered from a wayward tee shot on the final hole to sink his fifth birdie of the round and give himself some added insurance for a final round that will see not just McDowell and Havret breathing down his neck but Majors big guns Woods, Els and Mickelson.
He struck a confident note, though, as he looked ahead to the challenge.
"Today was a good day," Johnson said. "I hit the ball well.
"I had a couple of loose drives on the back, but I held it together pretty good and made a few good putts coming down the stretch.
"Nothing is going to change with the game plan tomorrow."