With just over one minute on the clock, Celtic opened the scoring when Ki Sung-Yong delivered a low free kickback into the St Johnstone penalty area from wide on the right and Stokes fired in from 18 yards.
Zilina tried to hit back and Robert Jez saw Cech spill his low free-kick before the keeper gathered at the second attempt, while Babatounde Bello had a header easily saved.
McGeady's stuttering run-up allowed him a fraction of additional time to place the spot-kick and, though McGregor got a hand to it, his lowkick found the net to bring the visitors right back into the game.