Derbyshire, who started the day 15 runs behind the Indians, declared their second innings at 156 for three after Billy Godleman top scored with 56 not out and Alex Hughes was unbeaten on 36.
Set a target of 142, the Indians made 143 for five after Murali Vijay top scored with 41 retired out and Ajinkya Rahane, who was also retired out on 39.
The biggest gain of the day was an impressive performance by the pacers. Bhuvneshwar Kumar was the pick of the lot with a fine spell of 6-3-5-1.
Pacers Ishant Sharma, Mohammed Shami and Stuart Binny also had a good outing with the bowl.
Brief Scores: Indians 341/6 dec. & 143/5. Derbyshire 326/5 dec. & 156/3 dec. Indians won by five wickets.