Fulton, 34, is unbeaten on 124 while Kane Williamson remains on 83 – his second consecutive half century – as the Black Caps went to stumps in Auckland at 250-1.
The only joy for England came for Steven Finn, who claimed Hamish Rutherford for 37.
This Test will decide a series winner after the first two were drawn in Dunedin and Wellington.
Although Kiwi captain Brendon McCullum admitted he would sent England in on the pitch due to its tinges of green, the surface was either lifeless or blunted by impressive Black Cap batting.
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