The lanky off-spinner justified his selection for the match with figures of 4-84 to restrict India to 219-7 on a spiteful pitch where the ball often kept low.
India's in-form opener Yashasvi Jaiswal (73) was done in by one such delivery and the hosts, who are 2-1 up in the five-match series, finished the day trailing England by 134 runs.
Dhruv Jurel (30) and Kuldeep Yadav (17) provided late resistance with a 42-run partnership for the unbroken eighth wicket.
"He was brilliant. He is a great young lad to have in the group," England's batting mainstay Joe Root said of Bashir.
"He should take a lot of confidence for the rest of this game and going forward as well.
"It looks like the pitch will keep getting worse. If we can get three early wickets, hopefully that puts us in a really strong position," Root said of the contest.
Earlier, Joe Root led England's remarkable turnaround from a precarious 112-5 with an unbeaten 122 before the tourists were all out for 353.
Tailender Ollie Robinson raced to his maiden test fifty, hitting a six and nine fours in his counter-attacking 58.
Ravindra Jadeja (4-67) removed Robinson and Bashir in the same over and went on to dismiss James Anderson to claim the final English wicket.
Root remained unbeaten after his 31st test hundred, a masterly knock studded with 10 fours.
India wobbled early in their reply after skipper Rohit Sharma, having made two, edged Anderson to be caught behind.
Jaiswal was fluent at the other end, though, and the nearest England got to dismissing him before his half-century was when Ben Foakes grabbed an edge. Replays, however, confirmed the ball had touched ground before the wicketkeeper collected it.
Shubman Gill (38) got off to a decent start but could not capitalise on it, falling lbw to Bashir.
Rajat Patidar (17) fell in the same manner to the spinner, while Jadeja (12) was snared at short leg off Bashir's bowling.
Jaiswal, the leading scorer of the series, passed the 50-mark for the fourth time in seven innings, which include two double hundreds.
The left-hander, however, fell just when he was looking good for another hundred and his exit put England in charge.
Jaiswal rocked bat wanting to play Bashir through the off-side but the ball stayed low and hit the toe-end of his bat before hitting the stumps.
India were reeling at 161-5 when left-arm spinner Tom Hartley twisted the knife dismissing Sarfaraz Khan and Ravichandran Ashwin.
Jurel and Kuldeep, however, frustrated England with their dour rearguard action.
