The home side made a strong reply to Hyderabad's first innings total of 434 all out with Bains, who hit 22 boundaries during his 302-ball knock, leading the charge on the third day at the HPCA Stadium here.
Bains got an able partner in Robin Bist (91 batting) and the duo stitched 132 runs for the fourth wicket that helped HP wriggle out of a difficult situation at one stage when C V Milind, the left-arm seamer, had three scalps with the home team score reading 195 for 3.
Hyderabad regained some ground late in the day as they removed the well-set Bains and Nikhil Gangta in successive overs.
But Rishi Dhawan, who needed 76 runs to achieve the double of 2000 runs and 200 first-class wickets, then halted the Hyderabad bowlers on their tracks with an unbeaten 48 in the company of Bist to give HP a chance to take a first innings lead.
Bist's 91 not out came from 154 balls and he has hit 14 fours so far.
Brief Scores:
Hyderabad: 434 (Tanmay Agarwal 118, Hanuma Vihari 101; Ronit More 4/84).