West Indies have a lead of 302 runs and could enforce the follow-on on Monday morning to keep the pressure on the tourists.
The Caribbean side declared their first innings at 484 for seven before lunch at Arnos Vale, in Saint Vincent.
Kraigg Brathwaite's marathon innings ended when he was caught at slip by Mominul Haque off Taijul Islam for 212.
Shivnarine Chanderpaul, who had been part of a 161 run parternship with Brathwaite, ended the innings unbeaten on 85.
The only consolation for Bangladesh was Islam ending with five wickets with his slow left-arm but things were soon to get worse for the tourists.
Faced with an hour to survive before lunch, Bangladesh quickly found themselves at 18 for two with Kemar Roach removing Tamimi Iqbal, who edged to Darren Bravo at second slip and Jerome Taylor removing Imrul Keyes in similar fashion.
Shamsur Rahman (35) and Mominul Haque (51) provided some much needed resistance as Bangladesh clawed their way to 105-4 but Benn was to run through the lower order.
Only skipper Mushfiqur Rahim, who carried his bat for 48, was able to cope with the turn and bounce generated by the Barbadian left-armer.
Benn ended with figures of 5-39 while off-spinner Jermaine Blackwood picked up 2-14 in his test debut.