Trott, who was 61 not out when England resumed on an overnight score of 118-0, reached 101 with 13 fours and one six as England declared its second innings on 254-6 to set an improbable target of 442 in less than two sessions of play.
England was left four wickets away from victory as Haryana was reduced to 133-6 in its second innings.
Seamer Tim Bresnan was the pick of the England bowlers, taking 2-13 in seven overs. Bresnan also had the tourists' best figures of 3-66 in the first innings.
England is looking to win its first test series in India in 28 years during the four-game series that commences Thursday at Ahmedabad. Mumbai, Kolkata and Nagpur are the other venues for the series.
For India, it is a chance to avenge a 4-0 defeat in a four-test series in England last year that led to it losing its No. 1 ranking.
After the four tests, England also plays two Twenty20 games before returning home for Christmas. It will then come back to play five one-dayers in January.
England XI 521 and 254-6 declared (Jonathan Trott 101 ret., Nick Compton 79, Ian Bell 48; Sanjay Budhwar 3-51) drew with Haryana 334 and 133-6 (Nitin Saini 50, Tim Bresnan 2-13)