The hosts outplayed their opponents to take full control of things, eventually being bowled out for 417 at tea before reducing England to 78-4 by stumps.
Bairstow, who fell for 15 runs late in the day, admits that a big effort is now required if England are to turn around the 56-run deficit and gain a foothold in the series.
"It's not the day that we were looking for, to be brutally honest," Sky Sports News quotes him as saying. "And obviously losing Ben Stokes in the last over just caps it off. We thought that if we got a couple of quick wickets this morning, we'd have been in a good position. That didn't happen and we're now fighting an uphill battle.
"It's something that we have to take on the chin, and we've still got a lot of batting to come, we have to remember that, and hopefully the guys can fight hard tomorrow. We'll keep digging in, like we did in the first innings, and like we did with the ball - we got ourselves back in a good position, but we just weren't able to capitalise today.
"It's still a pretty decent batting wicket. Obviously there are different ways of going about it, but you've got to try and look to be proactive. Their bowlers are very consistent - it was a good bit of bowling from Ashwin that got Moeen [Ali]. It wasn't the greatest of shots, but if he'd have gone through it, then it would probably have gone for six."