Trailing on first innings by 165 runs after the tourists were dismissed for 464 in the morning session, the home side go into the final day's play ahead by just 37 runs with eight wickets in hand.
First innings centurion Marlon Samuels, who generated considerable media attention for his saluting send-off of all-rounder Ben Stokes following his dismissal late on the third day, will resume on the last morning on 22 alongside Brathwaite.
However he returned to his cocoon of circumspection after the left-hander fell for 69 to a catch at the wicket off fast bowler Stuart Broad in the final session.
With Samuels calm and composed at the other end, the 22-year-old Barbadian duly completed his fourth Test hundred in the last nine matches, forcing seamer James Anderson through the off-side in the penultimate over of the day to reach the milestone off 228 deliveries with 11 fours.
"I didn't get a lot runs in the first few innings of this series so I'm really proud to make a contribution here," said Brathwaite.
Brathwaite and Samuels have so far put on 57 for the third-wicket following the entertaining stand between the opening batsman and Bravo.
They had come together just before the end of the morning session when Anderson dismissed local hero Devon Smith, bowled off the inside-edge for just two.
His early strike made amends in a small way for the careless manner in which he was run out to end the England innings and leave Joe Root unbeaten on 182.