The towering quick reduced England to 13-2 in pursuit of a mammoth 463 by removing Emilio Gay for 11 and then having Jacob Bethell lbw for a duck on the fourth day.
England were 54-3 at tea, still needing another 409 runs if they are to take an unassailable 2-0 lead in the three-match series.
Jamieson had interval figures of 2-10 in five overs.
Joe Root was 24 not out after the stand-in England captain became just the second batsman, following retired India great Sachin Tendulkar, to score 14,000 Test-match runs.
The scale of England’s task can be gauged by the fact that the highest fourth-innings total made to win in 149 years of Test history is West Indies’s 418-7 against Australia at St John’s in 2003.
The corresponding England record is the 378 they posted against India in Birmingham four years ago at the start of their dynamic “Bazball” era under head coach Brendon McCullum and captain Ben Stokes.
But this England side, showing five changes — including three debutants — from the one that won the first Test against New Zealand at Lord’s a fortnight ago, and with Stokes missing after he was dropped for breaking a team curfew, is a very different outfit from that of 2022.
The fourth over of their chase saw Gay clip Jamieson to midwicket.
New batsman Bethell came to the crease averaging a mere 8.75 in the first innings of Test cricket but a huge 72.5 in the second.
Those curious statistics counted for little, however, when four balls after Gay’s exit, he was hit plumb lbw for a duck to a Jamieson delivery that cut back into the left-hander.
England were now 13-2 and Root, not for the first time in his illustrious 165-Test career, walked in to bat with his side in dire straits.
Root, the immediate predecessor of Stokes as England captain, needed just two runs to join Tendulkar in the exclusive club of batsmen to have scored 14,000 in Tests and a quick single off Matt Henry took him to the landmark.
The 35-year-old, somewhat shyly given he was two not out, raised his bat as a large crowd in south London applauded his achievement.
Tendulkar’s 15,921 runs from 200 Tests could be Root’s next target.
But Root could only watch as Ben Duckett, on nine, recklessly tried to hook a Will O’Rourke delivery way above his head and spooned a simple catch to short midwicket.
Earlier, England eventually dismissed New Zealand for 362 in the tourists’ second innings, with Daryl Mitchell — dropped off Saturday’s first ball — making 68.
Henry Nicholls added just two runs to his overnight 119 not out, a century made after he was given the unenviable task of replacing Kane Williamson, with New Zealand’s greatest batsman stunning the cricket world by retiring from international cricket following England’s 115-run win at Lord’s.
New Zealand lost their last four wickets for 13 runs but any disappointment they may have felt at not setting England a target of more than 500 was replaced by the joy of Jamieson’s success. (AFP)
