West Indies, following on, were 76 for four at tea -- still needing a further 270 runs to make England bat again.
England were now just six wickets away from victory, with a minimum 38.4 overs left in the day.
Play could also potentially go on to 10:30pm local time (2130 GMT) if the umpires allowed England to claim the extra half hour in the belief they could win in that time.
Earlier Anderson, England's all-time leading wicket- taker, sparked a first-innings slump that saw the West Indies dismissed for just 168 -- 346 runs behind the hosts' 514 for eight declared.
Anderson finished with first-innings figures of three for 34 in 15 overs.
The Lancashire paceman took two for six in four overs in Saturday's first session, with the West Indies losing seven wickets for 89 runs in 23 overs before lunch -- all without the problems of batting under floodlights against the pink ball.
Joe Root, the England captain, enforced the follow-on and the wickets kept falling, with Kieran Powell out for 10 when Cook held a routine chance at first slip off Anderson.
West Indies were 42 for two when Test debutant Kyle Hope (12) was lbw to Toby Roland-Jones.
Shai Hope hung in for more than 30 minutes but then nicked Ben Stokes to second slip Root.
Brathwaite resisted for 107 minutes and hit seven fours before he went back to Birmingham-born off-spinner Ali and fell lbw.
Kyle Hope fell without adding to his overnight score when, off the sixth ball of the day, an Anderson delivery flew off a lenth to take the shoulder of his bat before carrying to Stokes at gully.
Five balls later, 45 for two became 47 for three when the ambling Powell was run out by a direct hit from Anderson, coming round quickly from mid-on.
The collapse continued when Anderson had Chase playing on for a duck.
Blackwood, however, drove first-change Roland-Jones for two superb straight fours.
Blackwood, on 28, was hit on the helmet when he failed to sway out of the path of a short ball from Stokes.
The gutsy Blackwood commendably cut Stokes's next ball for four.
He completed a 49-ball fifty including seven fours and he later drove Ali for six.
But Ali had Jason Holder, the West Indies captain, caught behind before Stuart Broad bowled Kemar Roach to leave the tourists 145 for eight at lunch.
Blackwood's two-hour innings saw him face 76 balls, with nine fours and two sixes.
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