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Published : 14 Aug 2016, 12:29 PM
The touring side won by 237 runs on the fifth and final day after bowling out West Indies for 108, the second lowest score by the hosts against India in the Caribbean.
The victory gave Kohli's men an unassailable 2-0 lead in the four-match series and ended a streak of six consecutive series defeats outside Asia.
Paceman Mohammed Shami was the chief destroyer with three for 15 while spinner Ravindra Jadeja and fast bowler Ishant Sharma took two wickets apiece.
"The credit goes to the whole squad, the way they have responded to the challenges," Kohli said at the presentation ceremony.
"We always aim to win matches, win series, especially away from India, and this has been a very good start."
The West Indies collapse was dismal apart from an assured 59 by number three Darren Bravo.
India had not won or drawn a series outside the sub-continent since their triumph in West Indies in 2011. They have now won three straight series in the Caribbean after losing seven of their first eight.
The touring side won the third Test despite losing a day to rain at the Darren Sammy Stadium in Gros Islet.
Anil Kumble took over as coach in June on a mission to improve the team's miserable form away from the sub-continent.
A 2-1 series victory in Sri Lanka last August was a promising sign but sceptics wondered if India could replicate that display on a different continent.
Positive culture
Man of the match Ravichandran Ashwin said a positive squad culture was an integral part of their success.
"Honesty is one thing that is really prevalent in this group," said all-rounder Ashwin who scored a first innings century.
"Nobody is worried about speaking their mind which is a very important thing in a team culture."
Defeat represented yet another setback for the Windies who have drawn two and lost eight of their last 10 Tests,
Captain Jason Holder had few complaints about the bowling but acknowledged major problems with his team's batting.
"The guys need to dig a little deeper," he said. "You can't win Test matches unless you put runs on the board.
"We've got to be a little more patient, put a bigger price on our wickets. I can't put my finger on exactly how I'm going to fix it."
Viv Richards, one of the Caribbean's all-time batting greats, said the fourth Test starting on Thursday could be crucial for several players.
"They will be humiliated in my opinion," he added. "There are a lot of guys in there who are fighting for their careers."
