The hosts already had the series in hand after chasing down 294 and 227 this week, and even after being a set a target of 322 on Thursday they still cruised to victory.
Bangladesh finally put up a score the pitch deserved, 5-321 with four half-centuries, but the hosts overcame a wobbly start with a controlled chase to reach 6-325 with four overs to spare.
The highest successful ODI run chase at Warner Park was also the third highest in West Indies ODI history.
Jangoo handled it like a seasoned pro, even though he came to bat in his first senior international with West Indies 4-86 in the 15th over.
Keacy Carty shepherded Jangoo and they combined for 132 runs until Carty fell for 95.
West Indies still needed 104 runs off 100 balls and Jangoo took up the reins. His unbeaten 104 off 83 balls was the second-highest score by a West Indian in an ODI debut after Desmond Haynes' 148 against Australia in 1978.
Nerveless in the 90s, Jangoo's fourth six, over long-on, gave him the hundred off 79 balls.
"It is surreal, an amazing feeling," Jangoo said. "I just did not want to make a duck on debut. He (Carty) is a close friend of mine, made it easy for me."
After Jangoo achieved his milestone, Gudakesh Motie finished off proceedings with 44 off 31 balls.
Earlier, Mahmudullah achieved his fifth fifty in six appearances at Warner Park, and fourth straight 50 at the ground for Bangladesh.
He and Jaker Ali took 104 runs off the last 10 overs, 59 off the last four. Mahmudullah's unbeaten 84 came off 63 balls, and Jaker's unbeaten 62 off 57.
"Three fifties in this series (by Mahmudullah) is a good moment for our team," captain Mehidy Hasan Miraz said. "We learn so many things from him."
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