After a goalless first half, Nigerian striker Dudu broke the deadlock in 78th minute at the Kalinga Stadium.
East Bengal now await the winners of the second semi-final between Mohun Bagan and Bengaluru FC.
It was a scrappy first-half which ended goalless.
East Bengal had the better chances with Syrian midfielder Mahmoud Al Amna missing the first real chance in the 19th minute, when he failed to tap in a cross from inside the box by Japanese midfielder Katsumi Yusa.
Dudu broke free from an Al Amna flicked header a minute later but with just the Goa keeper Kattimani to beat, hit straight at him.
Katsumi then missed two gilt-edged chances between the 33rd and 35th minutes and Goa got their best chance through Coro in added time, when his solo run found him in front of East Bengal goalie Ubaid, but he could not convert.
The half was highlighted by as many as four bookings, equally shared by both sides. Seriton and Edu Bedia went into referee Santhosh Bhaskaran's book for Goa while Cavin Lobo and Gurwinder received yellow cards for East Bengal.
East Bengal continued to press hard for a goal in the second half and Dudu and Katsumi again missed at least three golden opportunities to score between them, the Nigerian being the more profligate.
The deadlock was finally broken when another quality cross from the left flank by Katsumi found Dudu who tapped in past Kattimani.
While Samad Ali Mallik became the third East Bengal player to be booked, FC Goa captain Eduardo Bedia picked up his second yellow of the game for a lunge at substitute Kromah, who had come in for Dudu and Goa had to finish the game with 10-men on the field.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)