The seamer, playing his first match of the series, struck twice after England resumed the second day on 53-3.
Nightwatchman Craig Overton was the first to go, snared at first slip by India captain Virat Kohli.
Dawid Malan, who made 31, followed suit after Rohit Sharma at second slip dived to his right to take a sharp catch.
The Indian seamers, who had been bowling with a lot of discipline, suddenly took their foot off the pedal and boundaries started flowing.
Pope looked in particularly good shape and hit Shardul Thakur for three boundaries in the same over.
His aggression appeared to rub off on Bairstow who meted out similar treatment to Mohammed Siraj in the next over.
Siraj broke the 89-run stand after the lunch break trapping Bairstow lbw for 37.
Moeen made 35 before throwing away his wicket playing a rash stroke against Ravindra Jadeja.
The five-test series is level at 1-1.