India declared their whirlwind first innings on 285-9 from just 34.4 overs, after Bangladesh were bowled out for 233 on a whirlwind day four in Kanpur.
Bangladesh had to face a tricky 11 overs before stumps and reached 26-2, still 26 runs in arrears, after Ravichandran Ashwin trapped Zakir Hasan lbw for 10 and then bowled nightwatchman Hasan Mahmud for four.
Shadman Islam was on seven and Mominul Haque, who made an unbeaten 107 earlier in Bangladesh's first innings, was yet to score.
Left-hander Jaiswal smashed 72 off 51 balls including 12 fours and two sixes as India reached 100 off just 61 balls, breaking their own record for the fastest start to a Test innings.
It beat the 74 balls they faced to reach the mark against the West Indies last year.
"That was our message this morning, you know, with losing two days of the match... that positive intent was going to be key," India's bowling coach Morne Morkel told reporters.
"I thought the way the boys went up with the ball upfront was incredible to see. To show the intent with the bat was awesome," he added. "Our game plan was to see how can we get the winning result."
The former South African fast bowler said the crowds loved the batting show, which had "the T20 feel about it".
Jadeja 300 Test wickets
Jaiswal put on 55 off 24 balls for the first wicket with skipper Rohit Sharma, who was bowled by Mehidy for 23 in the fourth over.
Jaiswal kept up the onslaught, reaching his fifty off 31 balls before he was bowled by Hasan Mahmud.
But Virat Kohli, who hit 47 off 35 balls, and Rahul, whose 68 came of 43 balls, kept up the momentum.
Bangladesh spinners Shakib Al Hasan and Mehidy Hasan Miraz took four wickets each.
"They played to a plan and I was surprised," Mehidy said. "But after two to three overs we knew their plan and we played to our strength. Just tried to do some damage to their innings."
Former captain Shakib, 37, announced his international retirement on Thursday and could be playing his last Test if he is unable to return home for a scheduled South Africa series.
Earlier, Ravindra Jadeja, with his left-arm spin, took the final Bangladesh wicket of Khaled Ahmed, caught and bowled, to reach 300 Test wickets.
He became the seventh Indian bowler to reach 300 on a list headed by Anil Kumble with 619 Test victims.
"It's special," said Jadeja. "I've been playing Test cricket 10 years and I've finally reached this milestone. Feeling proud of myself."
Bangladesh had resumed at 107-3 under clear skies after rain and a soggy outfield had washed out the whole of Saturday and Sunday. Only 35 overs had been possible on Friday's opening day.
Left-hander Mominul, who began the day on 40, reached his 13th Test hundred before lunch with a boundary off Ashwin.
India lead the two-match series 1-0 after they won the opener in Chennai by 280 runs.