Ashwin finished with figures of 6-47 as India was set a 276-run target on a pitch with low bounce at the Ferozeshah Kotla Stadium.
The West Indies, enjoying a 95-run first innings lead, failed to build a big lead after resuming the day at 21-2 and was bowled out 72 minutes after lunch.
First-innings centurion Shivnarine Chanderpaul and captain Darren Sammy came close to half centuries as the tourists put up some resistance after being 84-7 at one stage.