India international Clifford Miranda bagged the other goal for Dempo while substitute Baljit Sahni netted a late consolation for East Bengal.
The visitors came into this game on the back of a four-match winless streak but they raced into an early lead when Miranda slotted home Billy Mehmet's low cross from the right flank.
Irishman Mehmet did brilliantly to get away from his marker and find Miranda, who just had to tap the ball in from six yards.
Backed by a full house in Kalyani, East Bengal tried to respond but only created half chances with both foreign strikers Edeh Chidi and James Moga going close.
But it was the Goan side who should have doubled their advantage and in an identical fashion as once again Mehmet did well to play in a low cross for Miranda but the Dempo skipper failed to connect to it.
Dempo almost immediately could have paid the price for that missed opportunity as East Bengal were awarded a penalty following Samir Naik's handball but Chidi uncharacteristically put his spotkick wide.
The Goan giants wasted another chance to increase their lead when Peter Carvalho failed to head Beto's well-measured corner despite being unmarked.
Arthur Papas's side finally got their second goal nine minutes after the restart when Beto found the back of the net from close range following a Miranda corner.
East Bengal goalkeeper Abhijit Mondal failed to collect the ball under pressure and it went towards the Brazilian, who was left with an easy finish.
At 2-0, East Bengal had a mountain to climb and their Brazilian coach Marcos Falopa introduced Sahni and Joaquim Abranches to get back into the contest.
But the AFC Cup semi-finalists still struggled to create clear cut opportunities with Subhashish Roywchowdhury only having to stop a deflected shot from Chidi.
East Bengal were throwing everything at Dempo but at the other end looked vulnerable to the counter attacks.
From one such break, Dempo won a penalty when substitute Joy Ferrao was brought down inside the box by Uga Okpara after a lovely pass from Jeje Lalpekhlua.
Okpara was sent off after receiving his second yellow while Beto successfully converted from the spot to seal the points.
Despite going down to ten men, East Bengal pulled one back through substitute Sahni, who found the back of the net with a shot from just outside the box.
But it only turned out to be a consolation strike as Dempo held on for their first win of this season to take their tally to six points from five matches.
East Bengal now have three points from two games and don't play until October 26 when they host Bengaluru FC.
Scorecard: Dempo 3 (Miranda 3, Beto 54, pen 76) beat East Bengal 1 (Sahni 85)
--IBNS (Posted on 17-10-2013)