Mashud's unbeaten 103 allowed Bangladesh skipper Habibul Bashar to declare for the first time in the Test side's short history and set Windies an improbable 336 to win from a maximum 29 overs.
Windies closed on 113 for no loss, Chris Gayle making 66 and Devon Smith 40.
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Bangladesh, with a lead of 158 but in trouble at 94-6 in their second innings at the end of the fourth day, batted determinedly to frustrate the home attack and snuff out any hopes Windies skipper Brian Lara harboured of a victory.
Rajin Saleh and Mashud took their seventh wicket stand to 44 before Saleh was trapped leg before by Fidel Edwards for 51.
Mashud, whose previous Test best was 61, and first innings centurion Mohammad Rafique added another 56 runs before Ramnaresh Sarwan ended the partnership soon after lunch. Wicketkeeper Ridley Jacobs claimed a catch at the second attempt to remove Rafique for 29.
Tapash Baisya (26) then gave Mashud solid support in a stand of 74 before Bangladesh declared on 271 for nine.
With no chance of reaching their victory target, the Windies openers cruised to a century partnership.
The second and final Test starts at Sabina Park, Jamaica on Friday.
Scorecard:
Bangladesh first innings 416
(Habibul Bashar 113, Mohammad Rafique 111, Mohammad Ashraful 81; P.Collins 4-83)
West Indies first innings 352
(C.Gayle 141, B.Lara 53; Mushfiqur Rahman 4-65)
Bangladesh second innings (overnight 94-6)
Hannan Sarkar b Edwards 9
Javed Omar c Jacobs b Collins 7
Habibul Bashar b Best 25
Rajin Saleh lbw b Edwards 51
Mohammad Ashraful c & b Sarwan 1
Faisal Hossain c Gayle b Sarwan 2
Mushfiqur Rahman lbw b Sarwan 0
Khaled Mashud not out 103
Mohammad Rafique c Jacobs b Sarwan 29
Tapash Baisya c & b Gayle 26
Tareq Aziz not out 1
Extras (lb-5 nb-12) 17
Total (for nine wickets declared, 105.2 overs) 271
Fall of wickets: 1-17, 2-21, 3-70, 4-73, 5-79, 6-79, 7-123, 8-179, 9-253
Bowling: Collins 17-5-42-1 (nb-3), Edwards 19-1-61-2 (nb-2), Lawson 16-0-60-0 (nb-7), Sarwan 20-9-37-4, Best 13-1-33-1, Gayle 19.2-7-33-1, Chanderpaul 1-1-0-0.
West Indies second innings
C.Gayle not out 66
Devon Smith not out 40
Extras (b-4 lb-1 nb-2) 7
Total (for no wicket, 23 overs) 113
Bowling: Tapash Baisya 3-0-26-0, Tareq Aziz 6-1-31-0 (nb-1), Mushfiqur Rahman 6-0-25-0, Mohammad Rafique 5-1-7-0 (nb-1), Mohammad Ashraful 3-0-19-0
Result: Match drawn.
Second Test: Kingston, Jamaica - June 4-8.
