The 23-year-old became the first person in history to record centuries in both innings of a first-class match at Headingley, but more importantly his 118 not out in trying circumstances ushered the tourists to an Investec series-levelling five-wicket triumph.
Beginning the evening needing 123 for victory as the floodlights punctured the gloom in West Yorkshire, Shai Hope easily survived a frivilous appeal after being struck on the pad by Broad.
They resumed on five without loss after England captain Joe Root had declared late on Monday with England 490 for eight in their second innings.
West Indies, hopeful at tea of pushing for a first test victory in England since 2000, was completely demoralized at stumps. The win, West Indies' first in a Test in England for 17 years, was all the more impressive as just over a week earlier they had been thrashed by an innings and 209 runs inside three days in the series opener at Edgbaston. We would have had a tiresome day with only one side going for a win.
Having seen Moeen Ali drop a sitter on the second day, Broad was then more than a little relieved to see Ben Stokes take a catch at fourth slip to remove Kieran Powell.
Straight As for 96 Altrincham Grammar School for Girls students
Notably, a significant 92 pupils achieved all A* and A grades, with 33 individual pupils securing 10 or more A* grades. The Art department saw 100 per cent A* to C results with one pupil not dropping a single mark for their A* grade.
You wonder whether Kraigg Brathwaite and Shai Hope knew that.
Roston Chase (30 off 58) chipped away at the target, frustrating England in obdurate fashion before launching Chris Woakes to mid-on where substitute fielder Mason Crane, only just on the park, grabbed a fine diving catch.
Nobody had ever made a century in both innings of a first-class match here at Headingley. An absolutely splendid game and well done to the West Indies. Moeen failed with an lbw review on Shai Hope, who made his century in 175 balls and was also dropped by Cook on 106.
Blackwood, who uppercut Broad for six over third man, tried to win the match in the grand manner.
Jermaine Blackwood was positive as he slashed his way to 41 off 45 balls to send the home team's fielders scurrying all over Headingley before he was out stumped with two runs still needed.
"(Hope) played exceptionally well - on the fifth day, against high-quality bowling, it was a great knock", said Root, whose two fifties in the match were eclipsed by Brathwaite and Hope.