After Monday's washout in Bristol England paid for a patchy performance with the ball as Suresh Raina hit a century in India's 304 for six.
Then England mustered only an insipid 161 all out in 38.1 overs, after a rain break between innings left a revised target of 295 in 47.
Debutant Alex Hales (40 from 63) and Cook (19 from 33) raised hopes in an opening stand of 54, but once first-change Mohammad Shami struck twice in four balls, the hosts had no answer, with Ravindra Jadeja finishing with figures of four for 28 from seven overs.
"We didn't play very well at all today," said the captain. "Credit to that partnership from MS [Dhoni] and Suresh Raina, they took the game away from us and we are quite slow to react to that. But even when we batted it wasn't a 160 wicket. You've still got to turn and play and to be brutally honest we didn't do that in any of our departments."
Chris Woakes was the pick of the England bowlers taking four for 52 but Chris Jordan had a torrid time as he recorded nought for 73 with 12 wides.
"I thought Chris Woakes bowled pretty well and obviously it's nice top see Alex Hales get up and running," added Cook. "What we have to do is dust ourselves down, come back in the next game and play better. There's a lot of talent in that dressing room to play better so it's not all doom and gloom. But if you play like that you don't win many games.
"CJ has had a really good start to his international career, unfortunately it's not always going to be a smooth ride for any of these young players and he'll learn from it. But he's got a great attitude to improving and he'll know that you can't keep bowling wides because it hurts the side. He'll know that more than anyone, he's an honest lad and when he gets it right he's a fantastic bowler."
India captain Dhoni was delighted to see his team get back to winning ways following comprehensive defeats in the final three Tests. "It's a very different format, it gives them the freedom to express themselves on the field," he said. "What it really does is bring them back into form."
South Africa chased down a 328 victory target to jolt Australia and get their campaign under way in the Zimbabwe Triangular Series with a seven-wicket win. Faf du Plessis, with his maiden one-day international hundred, and captain AB de Villiers, with his 18th, led South Africa to victory with 20 balls to spare.