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The Kiwis took the lead at the National Hockey Centre in the torrential rain when Simon Child finished high into the net to make it 1-0.
England looked to have levelled the bronze-medal match almost instantly, only for their goal to be ruled out after it was decided that Phil Roper's cross was played with the back of his stick, following a lengthy review.
But the 2010 bronze medallists were pegged back thanks to Alastair Brogdon's strike, after he reacted quickest to a loose ball and scrambled home the equaliser.
The game was turned on its head by Jackson who drag-flicked a thumping strike into the top corner from a penalty corner to give England the lead, although they could not keep hold of the advantage as New Zealand pulled back through Andy Hayward.
In what was a back-and-forth battle, Jackson found the net once more from another penalty corner, but, continuing the theme of the match, Child brought the scores level once more with his second of the game to take the contest into penalty flicks.
Both sides dispatched their first pens, but New Zealand failed with two of their next three to pave the way for Jackson to convert his flick to win the game for England.