West Indies, following-on, lost 19 wickets for 261 runs in the day's play as England surged to a 1-0 lead in this three-Test series.
Having been bowled out for just 168 in their first innings, after England had piled up 514 for eight declared on the back of man-of-the-match Alastair Cook's 243 and captain Joe Root's 136, the West Indies collapsed to 137 all out second time around.
Broad took three wickets for 34 runs in 10 overs, including a spell of three for four in 11 balls that saw him surpass England great Ian Botham's tally of 383 Test wickets.
Broad is now second in England's list of all-time leading Test wicket-takers, with only new-ball colleague James Anderson ahead of him.
"Very remarkable, very pleasing," Root told Sky Sports. "We talk about being ruthless. Stuart Broad has had a fantastic career... He has those spells in him where he can turn a game on its head.
"Today was a fine example of that."
Meanwhile Cook told BBC's Test Match Special: "It's always nice to win and contribute with a big score.
"We bowled well today -- Jimmy and Broady are world class."
'Windies must believe'
Defeat left the West Indies without a Test win in England since 2000.
Their squad is missing several star names, the legacy of a bitter dispute with their board and the attraction of the lucrative Caribbean Premier League Twenty20 tournament that is taking place at the same time as this tour.
But West Indies captain Jason Holder said his youthful side could not afford to lose heart in the short time between now and Friday's start of the second Test at Headingley.
"It is obviously very disappointing," Holder told Sky Sports.
"We have to just believe. It is not impossible, we can't drop our heads on one game, the series is not lost."
It was Anderson who did the initial damage in both West Indies innings.
Barely had a Windies first innings where Jermaine Blackwood's 79 not out accounted for nearly half the total finished, then Anderson struck again when Kieran Powell edged to former captain Cook at first slip.
West Indies were 42 for two when Test debutant Kyle Hope (12) was lbw to seamer Toby Roland-Jones.
Kraigg Brathwaite (40) resisted for 107 minutes before he was lbw on review to off-spinner Moeen Ali to leave West Indies 76 for four at tea.
Blackwood was then stumped by wicket-keeper Jonny Bairstow off Ali before Broad struck.
He had Roston Chase plumb lbw and Holder edged to Cook next ball.
Broad, who had drawn level with Botham, had a shot at an unprecedented third career Test hat-trick, but Kemar Roach survived.
Broad was soon celebrating again though, bowling Shane Dowrich to top Botham's mark as the former all-rounder looked on from a television commentary box.
Roland-Jones ended the match when he had Alzarri Joseph caught in the slips by all-rounder Ben Stokes.
The West Indies, 44 for one overnight, lost seven wickets for 89 runs in 23 overs in Saturday's first session as they subsided to 145 for eight at lunch.
Anderson had a wicket just six balls into the day's play when Kyle Hope was caught in the gully for his overnight 25.
The wickets kept tumbling with only Blackwood, who drove Roland-Jones for two stylish straight fours, offering any sort of resistance.
He showed plenty of character too when, after being hit on the helmet by Stokes, he cut the pace bowler's next ball for four.
Blackwood completed a 49-ball fifty and later hit both Ali and Broad for well-struck straight sixes.
The innings ended when last man Miguel Cummins was run out by Tom Westley's throw from midwicket as Blackwood tried to keep the strike.
First Published: August 20, 2017, 10:18 AM IST