Despite winning the toss, England only reached 268-8 and that gave India the platform to ease to an emphatic eight-wicket victory with over a day to spare.
After the setback, Cook had no excuses for his team's performance and admitted that England had been outplayed by their hosts.
The 31-year-old is quoted by BBC Sport as saying: "To get bowled out for 280 in the first innings, we knew we were behind the eight ball. We had a little bit of a sniff at 200-6, but credit to their lower order.
"It's been a frustrating four days. Credit to India - they outplayed us. We've got a few days off to go our separate ways."
England must now win the final two Test matches to save the best-of-five series.