England were 102 for five at lunch on the final day after resuming on 44 without loss in their second innings.
They needed a further 353 runs to reach what would be a Test record fourth innings victory total of 455.
More realistically, they had a minimum of 66 overs in which to hold out for a draw after winning the first Test of this two-match series by 124 runs at Lord's last week.
Trent Boult and Mark Craig took two wickets each in Monday's first session before part-time spinner Kane Williamson had Ben Stokes caught behind on the stroke of lunch.
No side have made more in the fourth innings to win a Test than the West Indies' 418 for seven against Australia at St John's, Antigua, in 2002/03.
England, after a rain-marred fourth day, resumed with Adam Lyth 24 not out and fellow left-handed opener Cook 18 not out.
But Lyth, who made a maiden Test century on his Yorkshire home ground in the first innings, fell for his overnight score when he was caught behind fending at a Boult away-swinger.
Off-spinner Craig then struck twice in three balls to reduce England to 62 for four.
Ian Bell had made just one when he turned Craig straight to leg-slip Williamson.
Bell, one of England's senior batsmen, has now managed just 55 runs in eight Test innings since his 143 against the West Indies in Antgua in April.
Joe Root had promised England would "come out all guns blazing" but he lasted just two balls before exiting for a duck when he turned Craig off the face of the bat only for the ball to hit Tom Latham on the chest at short leg before the fielder clung on to a very sharp chance.
