The West Indies were 206 for three in reply to England's first innings 258, a deficit of just 52 runs, at tea on the second day.
Brathwaite was 102 not out and Shai Hope unbeaten on 85.
Their unbroken stand was so far worth 171 runs.
The fourth-wicket duo had come together with West Indies in trouble at 35 for three after James Anderson had dismissed Kyle Hope, Shai's older brother, on his way to brilliant lunch figures of three wickets for 10 runs in 12 overs.
But not even Anderson could make further inroads as Brathwaite and Shai Hope batted through the second session.
Their partnership made England's decision to drop third seamer Toby Roland-Jones look all the more curious.
Roland-Jones had taken 14 wickets in three Tests at an average of under 20 apiece since making his debut in England's preceding 3-1 home series win over South Africa.
But despite those fine figures he was omitted at Headingley, with England recalling the fit-again Chris Woakes, ostensibly in a bid to give the Warwickshire all-rounder game time ahead of the upcoming Ashes tour of Australia.
The West Indies found runs hard to come by while Anderson was bowling under grey skies.
But after Anderson dismissed nightwatchman Devendra Bishoo, caught behind for his overnight one, and had Kyle Hope (three) well taken at second slip by England captain Joe Root, the tourists hit back when the swing specialist was given a breather.
- Decisions overturned -
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Brathwaite and Shai Hope were still in the middle when conditions for batting improved as the sun broke through.
Brathwaite, who resumed on 13 not out, successfully reviewed lbw decisions on 35 and 46, after first getting an inside edge to Stuart Broad and being outside off stump playing a shot when given out to off-spinner Moeen Ali.
With the very next ball after Ali's appeal was overturned, Brathwaite drove him for six to go to fifty.
After lunch, Shai Hope completed his fifty when he drove Woakes down the ground for his ninth four in 72 balls.
But he showed no sign of being shaken up.
He was, however, dropped on 72 when Mark Stoneman, back on his heels at short leg, couldn't hold on to a whip off Ali.
In desperation, Root gave the last over before tea to part-time spinner Tom Westley.
Having reached his fifty with a six, Brathwaite went to his hundred with another when he lofted Westley's penultimate ball before the interval high over long-on.
It was a stylish way for the 24-year-old to complete his sixth hundred in 39 Tests off 189 balls, also including 13 fours after over four-and-a-half hours at the crease.
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