Courtesy Ashwin's five for 67, India bowled out England for 255 in their first essay to take a commanding lead of 200.
"We have played him well to be honest. He bowled nicely today but the accuracy he bowls with is something that is a challenge. Every world class spinner has accuracy and that is probably his main asset, his pace variation is something that is going to be a challenge," Bairstow, who scored an impressive 53, said.
Bairstow and Ben Stokes (70) consolidated the innings, taking England to 190/6 from 80/5 at one a stage before eventually folding for 255 in reply to India's first innings total of 455. Ashwin struck in the second session of the day en-route to his maiden five-wicket haul against England.
At the close of play, India had already stretched their lead to 298 with two more days to go.
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"When we bat whenever it may be there's uneven bounce on the pitch. We will try to chase down whatever is set. We got to be still positive. At the end of the day, it's still the run chase. That's the exciting part of it however many that may be," he said about England's survival plan.
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"We are pretty happy with the way Ben and I played and then (Adil) Rashid chipped in. Coming at the end, I thought he played really nicely. We needed to show a little more fight in rest of the game be it in the first or fourth innings.
"The fight comes in many different ways, be it with the bat, ball or field. Me as a keeper I'm always looking for guys to be in right position. I think we showed some nice fight today and taking three in the evening session was a strong comeback within today's passage of play," said Bairstow.
"It was sore, don't get me wrong, but it was one of those things and didn't affect me too much. It was just a pain in the backside or rather the knee.