The left-arm spinner grabbed 6/94 in the first innings and 1/34 in the second as the West Indies finished at 100/2 after being set a 226-run target to win the match off 37 overs.
Opener Lendl Simmons made a 49-ball 44, while Kirk Edwards (28 not out) and Darren Bravo (24 not out) were the unbeaten batsmen when play was called off with 15 overs remaining.
No play was possible due to rain on Saturday and Sunday.
Sunny also became the third Bangladeshi to bag six wickets in an innings on debut in the morning session as the West Indies were bowled out for 244 in their first innings in reply to Bangladesh’s 350/9 declared.
The hosts declared their second innings closed at their tea total of 119/3, with left-handed Shahriar Nafees (50) scoring his sixth Test half-century with the help of eight fours.