Triplett, who opened the major with a 62 on Thursday, remained on top of the leaderboard with a 4-under-par 66 for a 54-hole total of 15-under 195. That's one shot better than Perry, whose third-round 67 left him right on Triplett's heels going into Sunday's final round.
Brandt Jobe was a distant third at 9-under 201 after a sterling 62 on Saturday that matched the tournament record. He shot a 29 on the back nine.
The highlights of Triplett's round were making a bunker shot for an eagle at No. 6 and chipping in for birdie at No. 9. He had only one bogey, at No. 7.
"I think what we probably both wanted to accomplish today was to get some cushion on the field," Triplett said. "I think (we were) trying to keep pace with each other more so than the guys behind us."
Perry fell out of a tie for the lead when he made bogey on the No. 18 after his approach landed in the rough around the green and he chipped past the hole. Triplett and Perry had started the day at 11 under, a tournament record for 36 holes.
A 30-foot birdie putt at No. 16 pulled Perry into a tie for the lead after he had fallen two shots off the pace.
"It felt like match play," Perry said. "The two of us have separated ourselves a little bit from the field and he kept putting pressure on. I'm going to have to really play good tomorrow to get ahead of him."
Bernhard Langer, bidding to win a third straight major, fell nine behind the leaders after a 72 on a windy day. He began the round only three behind the leaders.
Defending champion Gene Sauers missed the cut, but 67-year-old Tom Watson made it and finished the third round at 1 over.