Stokes cracked 78 not out from 57 balls to allow England to declare on 129-3, setting West Indies 312 to win or 85 overs to survive.
Broad took 3-1 on the fourth evening and tore through the top order with three more wickets on Monday to leave West Indies 37-4.
Their recovery came in the shape of a century partnership between Jermaine Blackwood and Shamarh Brooks, who both made fifties.
Stokes produced the breakthrough, having the tangled Blackwood top-edge a pull to a diving Jos Buttler in a sustained spell of short bowling.
Further resistance came from West Indies captain Jason Holder but, when he was bowled by Dom Bess, England could scent victory.