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Anthony Wolfe (7th minute) and Ansumana Kromah (34th) scored for Churchill while Chris Payne (67th) pulled one back for the hosts.
Churchill have won three matches on the trot since Pereira took charge. They had beaten Mohun Bagan and DSK Shivajians before Tuesday's match.
Despite the loss, East Bengal are still at the top of the table with 27 points from 13 matches while Churchill are now in sixth place with 15 from 13 outings.
The hosts were playing a high line but failed to pull it off in the initial exchanges due to their slow defenders as Wolfe shot a good chance wide.
The visitors did not have to wait much longer to nose ahead.
East Bengal's veteran defender Anwar Ali lost the ball with a lurking Wolfe stealing possession before a botched clearance by goalkeeper T.P. Rehenesh paved the way for the Trinidad & Tobago World Cupper to roll the ball into an empty net.
Against the likes of Brandon Fernandes, Chesterpaul Lyngdoh, Kromah and Wolfe who are all fleet footed footballers, the hosts' defence looked at sixes and sevens.
Four minutes past the half hour mark, Churchill doubled their tally after another spate of porous defending by the hosts.
Wolfe beat East Bengal's Ivan Bukenya for pace to set up Kormah who rounded the goalkeeper easily after dribbling past two defenders to glide the ball home.
To stem the rot, East Bengal coach Trevor Morgan introduced willis Plaza in place of the ineffective Mohammed Rafique in the 36th minute.
Morgan replaced the off-colour Bukenya with Arnab Mondal at the start of the second half with Lalrindika Ralte also coming on for Rowllin Borges ending all three substitutions for the red and gold brigade.
The efforts paid off to some degree as Chris Payne pegged one back for the hosts, scoring off a melee inside the box.
Churchill withstood sustained pressure from East Bengal in the closing stages to bag all three points and dent the hosts' hopes of winning the league after a gap of 13 years.