Rutherford scored 171 as New Zealand reached 402 for seven at the University Oval in reply to England's 167, giving them a lead of 235 with two days still to play.
Batting on his home ground Rutherford broke the record debut highest against England set by Australian Charles Bannerman who had scored 165 in the very first Test match of all, at Melbourne in 1877.
Rutherford told Sky Sports 1: "Just do exactly what you do in first-class cricket. That was the plan in Queenstown last week.
"It's tougher in this game with (Steve) Finn and (James) Anderson bowling."
Rutherford kept his celebrations low key after passing his hundred.
He added: "You are out there to do a job and the job is to score runs. The first hundred is very special."
Playing on his home ground, Rutherford knew what to expect.
"I know what the surface is doing and the surroundings as well."
Despite passing 150 he was disappointed to get out.
Rutherford added: "I wanted to be there for half an hour after the lunch break and then go from there. I don't really like getting out."
Bad weather, which had ensured the entire first day's play had been lost, returned to wipe most of the final session in this rain-shortened match.
James Anderson (four for 108) moved ever closer to the 300 club among his country's all-time leading Test wicket-takers, as the tourists tried to limit a lead which had nonetheless long put New Zealand in control.
It barely amounted to even the hint of a significant fightback, however, and did not change the fact England will have to bat for almost all the last two days here to stay level in the series after the first Test of three.
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