Paisley, seeking a first title on the tour, is 15 under-par for the tournament and leads home favorite Branden Grace (66).
He sank three birdies, but spoilt his card with a bogey on the par-four 16th, just his third dropped shot in 54 holes.
Grace, the world number 30 and highest ranked player in the field, finished strongly with five birdies on the back nine for a total of 14 under-par for the tournament.
He has eight European Tour career victories but has yet to lift his country’s Open championship, which is the second oldest in the world having first been formally played in 1903.
South African Jacques Blaauw (66), American Chase Koepka (68) and Frenchman Adrien Saddier (73) are a further two shots back.
Saddier was joint overnight leader with Paisley but after a hot-streak on the greens on Friday he could not find his range in the third round, two-putting 17 holes and three-putting the other.
Reporting by Nick Said; Editing by Ken Ferris