Sri Lanka, who suffered their first test loss to their South Asian rivals on Sunday, were bundled out for 234 in reply with nearly five overs remaining.
The hosts were reduced to 31 for three in the 11th over and never recovered.
Wicketkeeper-batsman Dinesh Chandimal's 59 was the only knock of substance from Sri Lanka's top order while all-rounder Thisara Perera struck a few late boundaries in his 55 from 35 balls.
Seamer Mustafizur Rahman was the most successful bowler for Bangladesh with figures of 3-56.
Earlier, Bangladesh lost opener Soumya Sarkar for 10 but recovered through a second-wicket stand of 90 between Tamim and Sabbir Rahman who scored 54.
Sabbir and captain Mushfiqur Rahim (one) were out in successive overs to bring Tamim and Shakib together and the duo rebuilt Bangladesh's innings before accelerating on.
Tamim hit 15 fours and a six in his innings while Shakib's knock included four boundaries and a six.
The second ODI in the three-match series will be played at the same venue on Tuesday.
(Reporting by Sudipto Ganguly in Mumbai; editing by Ed Osmond)
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