Saints had two good chances to go in front but Brendan Rodgers' side stepped up the pressure and came close on a number of occasions before Timothy Weah set up James Forrest, who got in front of Scott Tanser to score from close range after 78 minutes.
The visitors, who had Kristoffer Ajer sent off in stoppage time, sealed the win two minutes from time when a quick break ended with Callum McGregor picking out Weah to fire the ball past Zander Clark.
Next up for Celtic is a home game against Hibernian on Wednesday while St Johnstone, who stay sixth in the table, go to Hamilton Accies on the same night.
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