Goals from Fabio Pena (38'), Issac Vanlalsawma (65'), Bipin Singh (77') and Aser Dipanda (79') propelled the Meghalaya-based team while Amiri (89') grabbed a late consolation for the home team.
It was the home team which started the game on the right note when Vishal Kaith had to turn away a flicked header from Haroon Amiri after he latched on to a long ball from Abhishek Das in the sixth minute.
Amiri had caused several headaches for the Shillong defence before Thangboi Singto had had enough. He brought on Brazilian attacking midfielder Fabio Pena for Redeem Tlang who was anonymous on the right wing. The move eventually paid dividends for the men from Meghalaya in the 38th minute.
After Charles fouled Kinowaki in midfield, Issac sent in a brilliant delivery which was met by Fabio Pena, who lost his marker well and headed the ball past a stranded Karanjit Singh.
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It was the visitors who started the second half well, with Kinowaki and Issac dictating the play at will. It was the youngster from Mizoram who doubled Shillong's advantage in the 65th minute with an absolute peach of a free kick awarded after Echezona fouled Pena just outside the box.
Bipin Singh, who ran his heart out on the left wing, came up with the third goal in the 77th minute to effectively finish the game out when he was found in space just outside the box on a counter. The youngster went on to curl a sublime effort into the top corner, giving Karanjit absolutely no chance to save it.
Though Haroon Amiri got a goal back for the home side when he volleyed home after a scramble inside the Shillong box, the sheen off the away side's win was not to be taken away.
The massive win on the road sees Shillong Lajong leapfrog Bengaluru FC to the fourth spot with 16 points while Chennai remain rooted at the ninth position with 8 points.