England had resumed on 339-6 in an atmosphere of unremitting tension at The Oval on Monday, and a fired-up Siraj dismissed Jamie Smith and Jamie Overton before bowling Gus Atkinson to complete his five-wicket-haul.
After Prasidh Krishna had bowled Josh Tongue for nought, Chris Woakes walked out to bat wearing a sling to protect his dislocated shoulder with 17 runs still needed.
Atkinson hit Siraj for six to give England brief hope, and cleverly protected Woakes from the strike.
But player-of-the-match Siraj produced another brilliant yorker to remove Atkinson and earn India their narrowest Test win by runs.
India's victory allowed them to hold the Anderson-Tendulkar Trophy, freshly minted ahead of this summer, since they won the last series between these two sides.
"With 60, 70-odd runs to win with seven wickets in hand you don't get to see many games like this," India captain Shubman Gill said.
"Very happy to get this over the line. A little bit of luck for us."
Siraj was named player of the match after taking match figures of 9-190.
England, 3-301 at one stage, lost their last seven wickets for 66 runs, a collapse prompted by Harry Brook's reckless dismissal after he had made a superb century.
India suddenly had a chink of hope and they took full advantage, removing Jacob Bethell and Joe Root (105) before bad light and rain ended the fourth day early.
Resuming on Monday, England still needed 35 runs to complete their second-highest Test run chase and by far the largest for any team on this ground.
The Oval was full for the final act of a series that fluctuated wildly over seven weeks, and under grey skies one of the most dramatic Test endings duly played out.
"It has been one of the all-time best Test matches," said England coach Brendon McCullum.
"We turned up today with high hopes, but Mohammed Siraj and the passion of the Indian boys was too strong and ultimately they deserve to win this Test match," he added.
"We had our chances, we will look back on our dropped catches and the position we were in with the bat."
Siraj will have been relieved to finish as India's hero. On Sunday, he had had caught Brooke on just 19 only to step over the boundary rope, a reprieve that looked likely to cost his team the match.
Woakes was the not out batsman having not faced a ball but running bravely in obvious pain.
"I didn't expect him to come out like that, batting with one hand. Kudos to him," Gill said, after his team completed a lap of honour.
