Most of the field had to complete their second round on Sunday following the inclement weather on Saturday and 2011 champion Couples moved to 15 under par overall after 72 holes by carding 62 and 67.
Perry, however, managed to record back-to-back 63s - the latter of which was bogey-free - to move to 13 under overall at Fox Chapel Golf Club in Pittsburgh.
Duffy Waldorf is a further two shots back, while fellow Americans Mike Goodes and John Huston are on 10 under.
Colin Montgomerie's slim hopes were wiped out on Sunday after the Scot could only muster 70 and 68 to keep him at three under, which is currently good enough for tied for 16th.
Montgomerie will be hoping to push for a top 10-finish on Monday on his Champions Trophy debut.