Bangladesh lasted less than seven overs on Tuesday as the West Indies sealed their first home test win in more than two years.
Joseph (2-32) dismissed Hasan Mahmud (0), before ending Jaker Ali's innings on 31.
That left Bangladesh on 9-129, and Joseph then sent a fierce bouncer into the shoulder of Shoriful Islam. Shoriful tried to push on, but lasted just eight balls before he was forced to retire hurt, ensuring Bangladesh's innings came to an end and giving the home side victory.
It was the first time the West Indies have won a Test at home since they beat Bangladesh in June 2022, ending a four-match winless streak.
The West Indies declared at 9-450 in their first innings, but managed just 152 in their second.
Bangladesh declared at 9-269 in their first innings and finished with 132 in their second.
Justin Greaves was named man of the match after his unbeaten 115 in the West Indies' first innings. Greaves also took 2-34 off 11 overs in the tourists' first innings.
The second Test begins at Sabina Park in Kingston, Jamaica, on Saturday.
