The Indians successfully defended a target of 160, but Barnwell gave them some late jitters with fours and sixes in 34 not out from 16 balls in a last-ditch bid for a West Indies victory.
The visitors' spinners -- like Pakistan's before them -- squeezed the West Indies top-order batting, despite an attractive, run-a-ball 41 from Darren Bravo.
Harbhajan Singh was the most successful Indian bowler with for 25 from his allotted overs, conceding 77 along with fellow off-spinners Ravichandran Ashwin and Yusuf Pathan.
Because of India's stingy bowling, the home team -- like the visitors without lots of leading players -- could not overcome the rising scoring rate before a near capacity crowd that included embattled West Indies opener Chris Gayle, the leading scorer in the recent Indian Premier League.
"We played well, our bowlers, they executed well," said India captain Suresh Raina.
"We have lots of young players, who have completed well in domestic cricket, and they have a new coach, who has been speaking positive things, so they are looking forward to doing well.
"This was an important victory for us, and they are looking forward to the one-day matches now."
Subramaniam Badrinath stroked 43 to revive the visitors, and lead them to 159 for two, after Darren Sammy undermined their top-order.
Badrinath struck fours from 37 deliveries to breathe life in to the Indians' innings, after they stumbled to 63 for at the halfway stage.
The visitors' top order was reduced to rubble, after they were sent in to bat, when Sammy snagged for 16 from his allotted overs.
But Badrinath turned things around for India, with a stand of 61 for the fifth wicket with Rohit Sharma, and inspired controlled late-order hitting to beef-up the visitors' total, as they added 72 from the final overs.
Praveen Kumar gave a hint of things to come, when they started with a maiden over, and West Indies struggled through the opening overs.
Ashwin made the breakthrough in the fourth over, having Lendl Simmons dubiously caught at slip off Ashwin for nine in the fourth over -- TV replays proposing they never touched the ball.
Munaf Patel then bowled fellow opener Andre Fletcher for 11 in the next over, leaving West Indies 22 for.
The Indians met a roadblock, when Marlon Samuels joined Bravo and made 27, putting on 66 for the third wicket before Harbhajan removed both of them in the space of balls in the 16th over.
Harbhajan had Samuels caught behind off a top-edge, and bowled Bravo before Kumar had Sammy caught at long-on for a duck in the following over to leave West Indies 93 for. In the closing overs, the visitors kept their icy, as Barnwell brought the crowd to their feet with some vigorous blows, but it was all tiny, late.
"We allowed India to make twenty runs lots of, when they bowled," said Sammy.
"We require to return to the drawing board and work out our 'death bowling'. It did not go very well for us.
"When I went out for the toss, my spikes went right through the surface, so I knew there was a tiny moisture there, so it was a nice toss to win, and they decided to field, but they did not restrict them to the total they wanted.
"When they batted, they left ourselves much to do in the finish, but it was nice to see T20 players like Barnwell showing that if they are in a nice position, they can challenge totals."
The sides play the first One-day International in a series of on Monday at the same venue.
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