Broad boosted England to 369 before the hosts' seam attack caused problems for the visiting batsmen to leave the Windies 137/6 at stumps on day
Their partnership is so far worth 136 and it was vital in bailing England out of a troubled spot.
England were 131/4 at tea, with fast bowler Kemar Roach having removed second-Test century-makers Dom Sibley and Stokes.
Sibley had frustrated the West Indies with a near eight-hour hundred during England's 113-run win in the second Test at Old Trafford that levelled this
Ben Stokes - continued from where he left - was in top gear on the final day from the outset.
At the end of the day's play England were 37/2 with Ben Stokes (16*) and captain Joe Root (8*) in the middle.
Ben Stokes fell prey to an audacious reverse-sweep of pacer Kemar Roach - who finally got his first wicket of the series after 521 balls.
Stokes struck 176 – his second-highest Test score – as he put a tired West Indies attack to the sword, facing more balls than he
The three-Test series in England will mark the resumption of international cricket after it was suspended in March due to coronavirus pandemic.
Barbados-born Roach returned to training this week and the paceman made it clear he expects to be heading towards England soon.
Bracewell, who has played 125 first-class games, which include 27 Test matches for New Zealand, has pocketed 376 wickets.
Live Cricket Score IND vs WI, 2nd Test, Day 4: Live streaming of IND vs WI, live cricket score, IND vs WI live score, ball
West Indies require another 323 runs with only six wickets, if a physically indisposed Darren Bravo comes out to bat in the later stages.
Roach missed a hat-trick at Sabina Park but became West Indies' ninth-highest wicket-taker.
West Indies are 45/2 in their second innings, needing another 423 runs to level the series with six sessions to go.
IND 99/3 OVERS: 33.0
ENG 112 (48.4 Ovs)
IND trail by 13 runs
New Zealand vs Australia - 2nd T20I
New Zealand
Match begins at 06:30 IST (01:00 GMT)
04 Mar, 2021 09:30 IST
Narendra Modi Stadium, Motera, Ahmedabad
West Indies vs Sri Lanka - 1st T20I
04 Mar, 2021 03:30 IST
Coolidge Cricket Ground, Antigua
West Indies vs Sri Lanka - 2nd T20I
06 Mar, 2021 03:30 IST
Coolidge Cricket Ground, Antigua
West Indies vs Sri Lanka - 3rd T20I
08 Mar, 2021 03:30 IST
Coolidge Cricket Ground, Antigua
New Zealand vs Australia - 5th T20I
07 Mar, 2021 08:30 IST
Bay Oval, Mount Maunganui
New Zealand vs Australia - 3rd T20I
03 Mar, 2021 11:30 IST
Westpac Stadium, Wellington
New Zealand vs Australia - 4th T20I
05 Mar, 2021 11:30 IST
Eden Park, Auckland
Afghanistan vs Zimbabwe - 1st Test
02 Mar, 2021 11:30 IST
Sheikh Zayed Stadium, Abu Dhabi
Afghanistan vs Zimbabwe - 2nd Test
10 Mar, 2021 11:30 IST
Sheikh Zayed Stadium, Abu Dhabi
West Indies vs Sri Lanka - 1st ODI
10 Mar, 2021 19:00 IST
Sir Vivian Richards Stadium, Antigua
West Indies vs Sri Lanka - 2nd ODI
12 Mar, 2021 19:00 IST
Sir Vivian Richards Stadium, Antigua
West Indies vs Sri Lanka - 3rd ODI
14 Mar, 2021 23:00 IST
Sir Vivian Richards Stadium, Antigua
New Zealand vs Australia - 2nd T20I
25 Feb, 2021 06:30 IST | 01:00 GMT
University Oval, Dunedin
New Zealand vs Australia - 3rd T20I
03 Mar, 2021 11:30 IST | 06:00 GMT
Westpac Stadium, Wellington
04 Mar, 2021 09:30 IST | 04:00 GMT
Narendra Modi Stadium, Motera, Ahmedabad
New Zealand vs Australia - 4th T20I
05 Mar, 2021 11:30 IST | 06:00 GMT
Eden Park, Auckland
New Zealand vs Australia - 5th T20I
07 Mar, 2021 08:30 IST | 03:00 GMT
Bay Oval, Mount Maunganui
New Zealand vs Australia - 1st T20I
New Zealand beat Australia by 53 runs
India beat England by 317 runs
Pakistan vs South Africa - 3rd T20I
Pakistan beat South Africa by 4 wickets
Bangladesh vs West Indies - 2nd Test
West Indies beat Bangladesh by 17 runs
Pakistan vs South Africa - 2nd T20I
South Africa beat Pakistan by 6 wickets