The 52-year-old, winner of last month's Senior Players Championship, fired a six-under 64 in the third round at Omaha Country Club as leader Michael Allen stumbled to a 72 while hampered by a neck problem.
Allen had started the day with a five-shot lead, but he bogeyed two of the last three holes - his fourth and fifth mistakes of the round - to see his lead whittled down to two.
Perry applied pressure with two opening birdies before giving one back at the third, and he then eagled the sixth to vault into the top 10.
He dropped another shot at seven, but he got back on track with birdies at nine and 10 and he picked up further shots at the 16th and 17th to set the clubhouse target at six under.
Perry was joined in a share of second by Fred Funk, who recovered from a mid-round wobble with birdies at the last two holes to cap a 67.
Corey Pavin kept a bogey off his card as he matched Perry's 64 to leap into outright fourth on four under, while Rocco Mediate dropped four shots in five holes around the turn and laboured to a 72 to slip to three under.
Bernhard Langer (68) and Fred Couples (70) will both go into the final round on level par as only 17 players managed to get under the card after 54 holes.