It seemed India would take control of the match after Kevin Pietersen (6) departed with the score 94 for three but Trott and Ian Bell (24) put on an unbeaten 67-run stand to help their team gain a 165-run lead.
Earlier, India had declared at 326 for nine after batting for an hour in the morning.
With England opening up a sizeable lead, the match looks to be headed for a stalemate which is bad news for the hosts who are trailing the four-match series 1-2.
England captain Alastair Cook (13) failed to fire for the second innings running, caught behind off Ravichandran Ashwin.
Nick Compton (34) failed to take advantage of a good start and was trapped leg before wicket by Pragyan Ojha, who became the joint highest wicket-taker in the series with 20 wickets.
Ravindra Jadeja accounted for the dangerous Pietersen. The South African-born batsman failed to read the line of the ball and was castled by the left-arm spinner.
There were a few close shouts for the English batsman but Trott and Bell soon settled down taking away any momentum the Indian bowlers had built.
Earlier, India, who resumed the day at 297/8, put on another 29 runs in the first hour of the morning session before skipper Mahendra Singh Dhoni called in Ashwin (29 not out) and Ishant Sharma (2 not out) to the pavilion.
Ojha (3) was the only wicket to fall in the morning when he was clean bowled by left-arm spinner Monty Panesar, who picked up his first wicket of the match after bowling 50 overs.
Virat Kohli (103) and Dhoni (99) took India out of a critical position Saturday to bring the hosts at par with England.
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