The West Indies were 86 for two at lunch on the fifth day, still needing a further 236 runs in a minimum of 67 overs to reach an imposing victory target of 322.
Braithwaite, dropped on four, was 49 not out and Shai Hope 11 not out.
Stuart Broad may have been unfortunate in having Brathwaite dropped in the slips but, after removing Kieran Powell, his luck evened out with the paceman's freak run out of Kyle Hope, Shai's older brother.
But the first innings of this match saw Shai Hope, whose 147 was his maiden Test century, and Brathwaite (134) put on 246 in a total of 427.
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After rain delayed Tuesday's scheduled start by 15 minutes, West Indies resumed on five without loss.
James Anderson walked out just three wickets away from becoming the first England bowler to take 500 Test wickets.
Moeen Ali's 84 was one of six fifties in the innings with Root (72), Chris Woakes (61 no), Dawid Malan (61), Ben Stokes (58) and Mark Stoneman (52) also reaching the landmark.
England, who had been 169 runs behind on first innings, were helped considerably by several West Indies dropped catches, with the tourists' misses this match amounting to a total of 238 runs.
But the West Indies had an early reprieve Tuesday when normally reliable first slip Alastair Cook dropped Brathwaite, yet to add to his overnight four, off Broad as he failed to hold a two-handed, chest-high, chance to his left.
Instead it was Broad who struck twice in contrasting fashion as the West Indies' 46 for none became 53 for two.
Broad had a 'conventional' wicket when Powell (23) fended to Stokes at fourth slip.
He then removed Kyle Hope for the unluckiest of ducks when, having dropped a fiercely hit return catch from Brathwaite, the ball deflected off his thigh and into the stumps at the bowler's end, with the non-striker run out as he backed up.