England, 3-38 overnight, scored 376 at Edgbaston before making inroads into the West Indies top order with the wickets of captain Kraigg Brathwaite and the struggling Kirk McKenzie in the last 80 minutes of play.
Mikyle Louis (18 not out) and Alick Athanaze, unbeaten on five, will resume the Windies' second innings on Sunday, with the tourists trailing by 61 runs.
England took six balls to make a breakthrough with Chris Woakes bowling Brathwaite without scoring while McKenzie's poor form continued as he was caught behind off Gus Atkinson for eight.
Earlier, Jamie Smith and Joe Root both fell agonisingly short of a Test century while Ben Stokes and Woakes also scored half centuries on a day that started poorly for England but which they quickly turned around to dominate as they look for a clean sweep of the series.
Root's controlled innings took him past 12,000 Test runs, leapfrogging Caribbean great Brian Lara in the process.
Smith was bowled five runs short of a maiden test ton while Root scored 87 as England fought back from 5-54 to move ahead of the Windies' first innings total of 282.
Seamer Jayden Seales revealed Smith had been marked down as a "compulsive hooker" in the away dressing room but their attempts to expose a weakness fell flat.
Instead Smith responded with pulls and hooks, launching a huge six over the Eric Hollies Stand and cashing in after tea as the bumpers continued.
"I guess it's nice to have to set it out quite early on in your career, that you are someone who's going to go out and be positive, not afraid of opposition," Smith said.
"Moving forward, you're not just that sitting target. People are going to come and attack you and you're going to go and attack them.
"It might bring about my wicket a few times but I'm really not too worried about that because I'm out there to score runs. Pull shot, hook shot ... I'm always thinking about taking it on first."
The 24-year-old Smith, playing his third Test, looked a consummate veteran with an array of attacking shots as he sped towards the milestone, only to be undone by an off cutter from Shamar Joseph that stayed low and took out his stumps.
"If anyone said to me at the start of the day you've got 95, I'm definitely taking that," Smith said.
I'm a little bit gutted, I guess, and I'm sure I will be in a couple of days, looking back on missing out on only five runs. But milestones aside, it was great to put in a performance for team."
Smith came in after Root and Stokes put on a 115-run sixth-wicket partnership that hauled England back from a precarious position after they lost two early wickets.
