'Two-metre Peter', as 6ft 6in opener Fulton is also known, came of age as the second oldest New Zealander to post a century at the highest level for the first time.
The 34-year-old shared an unbroken stand of 171 with Kane Williamson (83 not out) in a stumps total of 250 for one - leaving England with scant chance already of fighting back well enough on a steadfastly benign pitch to break the 0-0 deadlock in this final Test of the series.
Fulton's share of the spoils was 124no from 277 balls, including 15 fours and three sixes, while Williamson was also an immovable force on the way past his own 118-ball half-century. England had only Steven Finn's breakthrough just before lunch, for the wicket of Hamish Rutherford, to show for their unequal struggle.
True bounce, a lack of pace and absence of lateral movement gave the tourists precious little to work with - derailing Cook's gamble that this drop-in pitch might be at its most fruitful for his bowlers on the first morning.
There was zero encouragement from the outset for England's front-line seam attack, under sunny skies, including first-change Finn at the same venue where he gave Rutherford a torrid time in the tourists' one-day international victory last month.
James Anderson twice beat Rutherford on the back foot, and then had Fulton edging high and just past third slip to go from 12 to 16.
But there were no clear-cut chances, apart from when Finn had Rutherford edging to Cook at slip on the back foot to end an opening partnership of 79.
Cook turned to Monty Panesar half an hour before lunch, only to see Rutherford hit the slow left-armer twice for six either side of the short straight boundary in his second over.
Fulton had already mis-hooked Stuart Broad for a steepling six almost straight over the wicketkeeper's head.
Nothing went England's way, before and after Rutherford's departure, and the compact and classy Williamson was convincing throughout.
Fulton passed his second fifty of the series with an on-driven four off Finn, and then accelerated past his previous best of 75 with a leg-side sequence of 4-4-6 in one Panesar over.
England did not bowl especially badly but were highly vulnerable in conditions which quickly limited their ambition.
There was further mitigation perhaps in the height differential between the Kiwis' second-wicket pair, Williamson almost a foot shorter than Fulton.
England's margin of error was therefore small, particularly given the peculiar dimensions of the playing area at this rugby-orientated stadium.
Anything short was fodder, given the easy pace, and if anyone over-pitched the shortest straight boundaries in international cricket gave them no protection.
Fulton, who had been out of Test cricket since late 2009 and began this series with an average in the low 20s, made a mockery of those questionable credentials.
He aimed leg-side more often than not, was strongest of all when the ball strayed full and on his pads, and was prepared to go over the top against Panesar.
When England retreated to damage limitation, with boundary sweepers and a long-on for their spinner, the opener lost impetus - having reached his century from 203 balls.
It was therefore Williamson who kept the Kiwis moving against the old ball.
For Cook, who had won his first toss of the series and only a second in nine attempts as England captain stretching back to his tour of Bangladesh as Andrew Strauss' deputy in 2010, there was to be no consolation.
Even the second new ball brought no respite, and the statisticians were scurrying to find worse first days in the field for England after winning the toss.
Their long search turned up alarmingly few examples.