Both the team played attacking game and the visitors drew the first blood from a penalty scored by Chhetri after a Sean Rooney shot from inside the box hit defender Pawan Kumar's hand in the 25th minute at Tilak Stadium.
Salgaocar fought back and restored parity in the 32nd minute when Gurjinder's dipping free kick from almost 35 yards bounced awkwardly and beat keeper Lalthuammanwia Ralte to find the net.
But the visitors' joy was shortlived as Douhou Pierre fired one in from the top of the box, giving the rival keeper no chance to react in the next minute.
Chhetri than turned the provider when he set up Rooney on the left, who placed the ball into the net to the left of keeper Subrata Paul at the stroke of half time which ultimately proved to be the winner.
Bengaluru FC play their next match against Mohun Bagan on December 30, while Salgaocar take on Shillong Lajong on December 31.