The tourists were building on their strongest position of the series, granted by Mark Wood's inspired display of pace bowling, and resumed 142 ahead with all 10 wickets intact.
Denly was badly dropped on 12 off Shannon Gabriel but used his reprieve to show his potential, unfurling a series of pleasing drives. The 32-year-old hit 11 boundaries and scored at a consistently lively rate before wafting at Gabriel and nicking behind.
Joe Root (45no) and Jos Buttler (37no) came together in a fourth-wicket stand worth 60 as they milked the gentle spin of Kraigg Brathwaite and Roston Chase.
The West Indies were a bowler down for most of the day, Keemo Paul leaving the field on a stretcher after straining a quadricep pursuing one of Denly's cover drives, and lacked the vibrancy that propelled them to victory in the first two Tests.
Having safely negotiated a 10-over spell on the second evening, England's fragile opening partnership was parted by the first ball of the day, Burns gently flicking Paul's innocuous effort to square leg. A score of 10 lowered his average to exactly 25 from six Test appearances, though he is likely to get further chances at home in the summer.
The same is probably not true of Jennings, who clung on for 99 balls and 23 runs but never looked at ease and was ultimately bowled in ugly, slightly unfortunate fashion after Alzarri Joseph's delivery took a detour off his thigh pad. He averages 0.19 more than Burns from his 17 caps, and has two centuries to his name, but at present he does not have the confidence or tempo to continue at this level.
The West Indies should also have sent the third member of England's top three back cheaply when Denly fended Gabriel to third slip off the shoulder of the bat but Shimron Hetmyer bungled the chance.
Denly was quick to settle after that, catching the eye with a couple of sweetly-timed cover drives and finding regular gaps behind square.
He reached his first Test fifty in 74 balls but missed the chance to make a defining score when he tried once too often to carve Gabriel through the off side.
Root and Buttler were mostly untroubled as they scored comfortably against a lethargic attack.