Celtic had forged ahead in the first half but were pegged back by St Johnstone three minutes past the stoppage in the second half as Brown’s side thought they had snatched a point from the Scottish champions.
Nonetheless, Giorgios Giakoumakis had other plans as he handed Celtic the three points with a 95th-minute winner.
Brown, who has two international caps for Malta, was in the starting formation as he played his ninth game for St Johnstone in the 2022/2023 campaign so far.