Chhetri and Singh scored in the 28th minute and extra time of the match through penalties, respectively, while Gilbert Oliveira scored the lone goal for the visitors in 85th minute after getting a penalty.
The first half saw an even-steven contest in front of the 7,700 spectators, but Salgaocar possessed the ball most of the time.
In first 28 minutes of the game, the visitors spilled many half chances they could have converted into goals, but their attempts were repulsed by ever-alert Bengaluru goalkeeper Pawan Kumar.
But after they managed to score their first goal, Bengaluru came into their own and played some magnificent football, nearly striking thrice at the visitors' post.
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Compared to the first half, the second half was drab and not so interesting with both sides making half-hearted attempts to score from the shooting range.
Fifteen minutes before the long whistle, Salgaocar lost the plot, but were lucky to convert the penalty.
They however, conceded another penalty to the hosts, who did not fail to seal the match.
Two attempts by Salgaocar's Mathew Foschine and Frances Fernandes within 20 minutes added pressure on the hosts even as Pawan saved their strikes outside of the box to put Bengaluru defense in disarray.