The hosts went past Afghanistan's 278-3 against Ireland in 2019. The Indian total is the second highest in all T20 internationals after Nepal's 314-3 against Mongolia at last year's Asian Games in China.
Samson hit 111 of 47 balls -- his first T20 international ton -- studded with 11 fours and eight sixes after India elected to bat first in Hyderabad.
In reply, Bangladesh managed 164-7 to go down by 133 runs in what was former skipper Mahmudullah Riyad's last T20 for the nation.
Mahmudullah made eight before he fell to speedster Mayank Yadav and walked off with a smile and pat on the back by India skipper Suryakumar Yadav.
"We achieved a lot as a team," said Suryakumar. "I want to have selfless cricketers. We want to enjoy each others' success. We are having some fun."
Najmul Hossain Shanto's Bangladesh, who came on the India tour at the back of a two-Test sweep in Pakistan, were outplayed by the hosts as they lost the earlier Test series 2-0.
"We didn't play our best cricket," said Najmul. "We didn't execute our plans as a batting unit in all the three matches. We should believe in ourselves that we can compete."
Samson, a wicketkeeper-batsman, stood out and named player of the match as he hammered leg-spinner Rishad Hossain for five straight sixes in an over en route to his ton in 40 balls.
He put on 173 runs for the second wicket with Suryakumar, who hit 75 off 35 balls but his knock was overshadowed by his partner's blitz and the mammoth total.
Samson finally fell to Mustafizur Rahman and walked back to a standing ovation by the weekend crowd.
Suryakumar fell in the next over by Mahmudullah, who announced his retirement from the format ahead of the second match.
But there was more punishment in store for Bangladesh as new batters Hardik Pandya, who struck 47 off 18 balls and Riyan Parag, who made a 13-ball 34, kept up the charge with regular boundaries.
India surpassed their previous best total of 260-5 against Sri Lanka in 2017 as they scored at nearly 15 runs an over.
Bangladesh were never in the chase but wicketkeeper-batsman Litton Das, who hit 42, and Towhid Hridoy, unbeaten on 63, attempted to put up a fight.
Leg-spinner Ravi Bishnoi took three wickets including Litton and Najmul in his four overs.
Pandya was named player of the series for his 118 runs, one wicket and five catches in the three matches.
India's Test team, led by Rohit Sharma, will now host New Zealand in a three-match series starting next week with the opener in Bengaluru.
