The visitors were also helped by some poor fielding from the hosts, who dropped six catches in all during India's second innings.
Resuming at the Oval on Saturday on 75/2 with a slender lead of 52, India needed a strong day at the crease if they were going to keep their hopes of leveling the series alive.
Jaiswal resumed on 51 not out alongside nightwatchman Akash Deep and provided just what the doctor ordered - though he certainly rode his luck after being dropped no less than three times.
England would have been desperate to at least dismiss Deep early but it wasn't to be, as the seamer proved to be an excellent foil for Jaiswal. Together, the duo combined for a 107-run partnership before Deep was finally caught off a leading edge shortly before lunch, having just made his maiden half-century.
India's lead was building, however, and while neither Shubman Gill (11) or Karun Nair (17) lasted very long after the break, Jaiswal went on to make his sixth Test century, eventually departing for a well-played 118 off 164 balls.
England would have been hoping to wrap up India's lower order in quick fashion after Jaiswal's dismissal, but Ravindra Jadeja, Dhruv Jurel and Washington Sundar had other ideas.
Jadeja and Sundar would both make half-centuries at a rapid pace while Jurel added 34 as England's bowling attack wilted late on, with Sundar's 53 off just 47 at the end of the innings in particular rubbing salt in the wounds.
By the time the visitors were finally dismissed for 396 they had amassed an imposing lead of 373, with Josh Tongue's 5/125 the only real bright spark for England in the field.
The home side looked to be gaining back some ground at the end of the day as openers Ben Duckett and Zak Crawley put on a 50-run partnership without loss, but with just a few balls remaining in the day and the safety of the pavilion within reach, Mohammed Siraj struck, dismissing Crawley with a superb yorker to make it advantage India heading into day four.
At the close of the second day of the fifth Test at The Oval, India held a slender 52-run lead at 75-2, capitalizing on England's costly fielding lapses to take control of this series decider.
On a rain-interrupted opening day at The Oval, England clawed their way to a strong position, reducing India to 204-6 in 64 overs, but their joy was tempered by a worrying injury to key seamer Chris Woakes.
Ben Stokes says he is ready to push through the pain barrier once more as England prepare for a decisive fifth Test against India at The Oval.
Joe Root became the second highest run-scorer in Test history to lead England to a commanding position at the close of play on the third day of the fourth Test against India in Manchester.
