The home side came agonisingly close to becoming the new Asian champions with nine needed in the final over. However, Cheema conceded only six runs and took a wicket in the process to see his side home.
Shahadat Hossain took the strike in the final ball after Cheema bowled Abdur Razzak when a boundary would have done it for Bangladesh. But he could manage just a leg bye as the home side were restricted to 234 for eight in reply to Pakistan's 236 for nine.
Tamim Iqbal scored 60 off 68 balls, his fourth half-century in as many matches in the tournament, and Shakib Al Hassan (68) added 89 runs for the fourth wicket stand with Nasir Hossain (28) to put Bangladesh in control.
Umar Gul broke their resistance, inducing Nasir to hit a catch to Misbah-ul-Haq at mid-wicket and Cheema silenced the roaring Sher-e-Bangla crowd in the next over when he bowled Shakib around his legs.
Skipper Mushfiqur Rahim was caught by Nasir Jamshed on the square-leg boundary off Cheema for 10 then Mashrafe Mortaza struck 18 off eight balls.
Shahid Afridi won the man-of-the-match award after scoring 32 runs off 22 balls and conceding 28 from his 10 overs.
Pakistan opener Mohammad Hafeez scored 40 off 87 balls and wicketkeeper Sarfraz Ahmed contributed 46 not out.
Sarfraz took 19 runs off the final over bowled by Shahadat that finally proved to have made the difference Shakib, who also took two wickets for Bangladesh alongside Mortaza and Razzak, was adjudged player of the tournament.
