Left-handed batsman Jaiswal added four to his overnight score of 76 with a four, but was out after bowling his fourth ball of the day towards spinner Root, who jumped high to take the catch.
Rahul and Gill, who started the day on 14, attempted to rebuild a 36-run partnership but Hartley made the breakthrough with his left-arm spin.
Hartley had a tragic introduction to Test cricket when Jaiswal welcomed him with a six off the very first ball. He gave away 63 runs in nine overs on the first day.
Due to a loose shot from Gill, he was caught at midwicket for 23 runs.
England played their aggressive “baseball” brand of cricket after opting to bat early in the five-match series and scored 246 runs thanks to a 70 from captain Ben Stokes.