Mohamed Irshad scored a goal deep into injury time but that didn't matter in the end as Khalid Jamil's side remained on 10 points from 16 games. Kerala now have 23 points from 13 outings.
At the half-hour mark, East Bengal appealed for a penalty, but a freekick was awarded at the edge of the box instead. Silva's strike from the set piece was punched clear by junior World Cupper Dheeraj Moirangthem.
The win ended BFC's five-match winless run as they jumped from 10th to eighth position, three points behind Chennaiyin FC.
Kerala have dropped to fifth with 23 points from 14 games.
Adrian Luna's stunning goal was unable to earn Kerala Blasters victory as FC Goa fought back in a thrilling Indian Super League match which ended 2-2, at the Tilak Maidan Stadium, in Vasco, Goa, on Sunday.
For Mumbai, Diego Mauricio scored a consolation goal from the spot in the 71st minute.
There was more misery for the league leaders after half time and Vazquez produced a moment of magic to start the fireworks.
The Gaurs ended their campaign at ninth place with 19 points. Kerala finished fourth with 34 points and eagerly await to learn their semi-final opponents, one between Jamshedpur FC and ATK Mohun Bagan.
League Shield winners Jamshedpur looked off-colour upfront with Daniel Chima Chukwu spurning two gilt-edged chances in the first half and Greg Stewart also missing his usual trickery in the final third.
ATK Mohun Bagan moved to the top of the table and Kerala Blasters FC entered the top-four after the two sides played out an entertaining 2-2 draw in the Indian Super League.
It was Kattimani's day as his heroics helped Hyderabad prevail in the shoot-out. He saved the spot kicks from Mark Leskovic, Nishu Kumar and Jeakson Singh.
Barcelona slipped five points behind La Liga leaders Real Madrid when Alvaro Vazquez netted a shock late equaliser to snatch a 1-1 draw for Espanyol in the Catalan derby on Sunday.
A Gareth Bale header and a Cristiano Ronaldo penalty gave Real Madrid a much-needed 2-0 victory over Eibar in La Liga on Sunday, following their hammering by leaders Barcelona last weekend, and put them six points off the top of the standings.
Real Madrid moved four points clear of Barcelona at the top of La Liga with Cristiano Ronaldo scoring either side of a Gareth Bale strike in a 3-0 win at Getafe on Sunday.
Valencia's Paco Alcacer, Dani Parejo and Pablo Piatti found the target as they took full advantage of the first-half dismissal of Malaga's Sergi Darder to win 3-0 at the Mestalla on Friday.
New Valencia coach Gary Neville remains without a La Liga win after his side twice recovered from a goal down to rescue a fortunate 2-2 draw at home to Getafe on Saturday.
Real Madrid ensured top spot in La Liga at the midway point of the season when Cristiano Ronaldo struck twice in a 3-0 win at Getafe on Sunday before Lionel Messi's hat-trick helped Barcelona to a 4-0 victory at Deportivo La Coruna.