The left-hander picked up 11 strokes through 15 holes, only to bogey his final two holes for a nine-under 62.
The somewhat deflating finish left Potter, whose lone PGA Tour victory was in 2012, locked with fellow American Johnson at 14-under 201, two strokes clear of Australian Jason Day and American Troy Merritt.
The leaders will be able to keep an eye on each other on Sunday when everyone plays the final round on the Pebble Beach host course.
Johnson missed a chance to put some distance between himself and the pack when he negotiated the inward half in two-over, but seemed unfazed.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"The couple of bogeys I made the wind was a little bit tricky and I just got on wrong side of the holes, which I knew better," he said in a greenside interview.
"All in all felt good about the game, feeling really good about tomorrow."
Johnson, who won the Tournament of Champions by nine strokes in January, is chasing a third Pebble Beach victory, after previous successes in 2009 and 2010.
Rory McIlroy missed the 54-hole cut as he finished at one-under, but he almost went out in spectacular fashion with what would have been an astonishing eagle at the par-five 18th.
After pulling his drive into a water hazard -- the Pacific Ocean -- McIlroy took a penalty stroke and then hit a brilliant third shot from 220 yards.
His ball clattered against the pin but did not drop, a birdie only minor consolation.
The former world number one, however, did not sound too worried about the state of his game.
"I just need a few putts to fall," he said.
"I let the greens yesterday get into my head and it was hard to get out of that mindset."
(Reporting by Andrew Both in Adelaide, South Australia; Editing by Greg Stutchbury)
