The Indian captain reserved his best for the big day as he struck in the 15th, 65th and 89th minute to seal the issue for the ISL debutants.
For Pune City, Jonatan Luca scored through a brilliant free-kick in the 82nd minute to raise visions of a comeback which didn't happen in the end.
An unmarked Chhetri opened the deadlock 15th minute after a nimble-footed Udanta Singh kept a neat cross on the far post.
Dimas Delgado essayed a neat lob for Chhetri, who raced past a helpless Rohit Kumar before slotting past Kaith from the edge of the box to mark his 13th of the tournament in 89th minute.
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Pune also had a couple of chances, before Bengaluru could settle down into the match. They had a wonderful opportunity in the 30th minute as Emiliano Alfaro picked out Marcelinho's run into the box on the left. The forward flashed a cross across the face of the goal from the byline, but there was nobody to meet it.
The rivals also missed a very close chance in 48th minute as an unmarked Alfaro at the far post headed over the bar after Golui sprinted down the wide right to keep a measured centre.
Pune had their chance in sixth minute as defender Sahil Panwar shot over the bar after Alfaro controlled the ball inside the box off a corner before laying the ball off.
Pune also showed their desperation to get into the match after conceding the goal, as they received three yellow cards including Diego Carlos, who indulged in the most disgraceful acts.
He fouled Udanta, but refused to go away after the referee ordered him to go away. Sahil and Rafael Lpez Gmez were other players yellow carded.
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