Chasing 447 for victory, Sri Lanka were bowled out for 208 in the second session after skipper Dimuth Karunaratne's valiant 107, on a pitch that turned from day one.
Fast bowler Bumrah ended Karunaratne's spell and got one more to take his match tally to eight wickets while spinner Ravichandran Ashwin took four over the day.
Shreyas Iyer and Rishabh Pant, respectively named man of the match and man of the series, scored 67 and 50 in India's 303-9 declared on day two.
Their key roles brought an emphatic victory that left the hosts unbeaten in all three of their home day-night Tests and extended their streak at home to 15 series wins in the five-day format.
India also won the opening Test in three days and this win gives new all-format captain Rohit Sharma a winning start to his Test leadership.
"It was a big thing to lead in Tests," Rohit told reporters after the win.
"I had the support of my teammates. They were telling me what to do and what not to do. There are a few senior members in the team who understand the game well and (I) had their inputs as well."
Left-handed Karunaratne made a second wicket stand of 97 before Ashwin broke through for the stumping of the captain's overnight partner Kusal Mendis, who recorded his 12th Test half-century.
Karunaratne went on to wage a lone battle and raised his century -- the sole hundred for a Sri Lankan batsman in this Test series -- with a boundary off Bumrah.
"I would have been more happy if we had won," Karunaratne said afterwards.
"We batted under lights and it was tough," he added. "We have a good team, (but) the batting didn't capitalise on these conditions."
Lakmal farewell
India, who gained crucial Test championship points in the whitewash, remained dominant from day one after Iyer hit 92 and lifted the team from a precarious 86-4 to 252 all out.
Iyer was named man of the match while the left-handed Pant was crowned player of the series for his 185 runs in the series and wicketkeeping duties.
Bumrah then returned figures of 5-24 for his maiden five-wicket haul at home to dismiss Sri Lanka for 109.
"Bumrah's performance was magnificent without a doubt," Rohit said of his deputy.
"To come out and bowl like that in these kinds of conditions shows how much skill and ability he has."
The series will be remembered as marking Virat Kohli's 100th Test in the opening game in Mohali, though the star batsman scored just 45, 23 and 13 in the three Indian innings.
The former captain twice fell lbw in this Test to take his average down to below 50 for the first time since 2017, but the crowd at M. Chinnaswamy Stadium gave him a rousing reception.
Sri Lankan quick Suranga Lakmal took one wicket in his last Test and finished his career with 171 scalps in 70 five-day matches since his 2010 debut.
Lakmal scored one before being bowled by Bumrah, who rushed to bid farewell to the 35-year-old veteran with an arm around him and the Indian team joined in.
The tourists, who lost the opening Test by an innings and 222 runs, suffered a total whitewash after they lost the preceding Twenty20 series 3-0.
Players from India and across the world will now assemble for the Indian Premier League, which begins March 26.
