Broad, who had taken four wickets in the tourists’ first innings in the morning, continued by removing opener John Campbell without scoring and then nightwatchman Kemar Roach (4) after the West Indies were put in for the last six overs of the day.
England had declared on 226 for two in their second innings as Rory Burns scored 90, Joe Root a quickfire 68 and Dom Sibley 56.
They accelerated the run rate in yesterday’s evening session, seeking to build a large lead but still leave themselves enough time to bowl the West Indies out again and win the Test.
England were hastened by the prospect of today’s play being heavily curtailed by rain, leaving them the last day’s play tomorrow to try engineer victory and take the series 2-1.
Burns had to sacrifice the personal milestone of a Test century to allow England a chance to bowl at the tourists in the late afternoon gloom, in search of an early inroad.
He abandoned the studied approach that got him to the 80s to slash at the ball and was out when he skied spinner Roston Chase straight up into the air and caught.
Root was not out, scoring his runs off just 56 balls as he went after the bowling with characteristic flair.
Sibley was the first England wicket to fall after a 114-run partnership with Burns, trapped leg before wicket by Windies captain Jason Holder.
England began their second innings just before lunch after Broad finished with figures of 6-31 in 14 overs, mopping up the tail to dismiss the West Indies for 197 in the first innings and give his side a 172-run lead.
But it took a frustrating 54 minutes for England to make a breakthrough after the West Indies had resumed on 137-6, facing the possibility of being asked to bat again.
Holder led the fightback as he scored an important 46 runs to help his side avoid the follow-on before Broad had him lbw.
The series is level after West Indies won by four wickets at Southampton before England secured a 113-run victory at Old Trafford.