Rutherford sailed to 167 not out - matching England's first innings team total - as New Zealand went to lunch at 2-255, giving them an overall lead of 88.
He was out straight after the lunch break for a magnificent 171.
The 23-year-old left-hander toyed with the English attack, even after losing opening partner Peter Fulton for 55 when the score was 158, the pair having added 27 from their overnight score.
The morning session belonged to Rutherford, who scored 90 of the 124 runs, accelerating quickly from his overnight score of 77.
It took him 138 balls to reach his hundred, becoming the ninth New Zealand centurion on debut.
His score is second only to Mathew Sinclair's first-up 214 against the West Indies at Wellington in 1999.
The ton was raised with a sumptuous cover drive to the boundary, the trademark shot of an innings that has featured 22 fours and three sixes.
Two of the sixes came in the space of three balls from Panesar as Rutherford passed the 150 mark, just the 21st player from any country to achieve the feat on debut.
England, perhaps stung by chastisement from their press after a reckless day two batting performance, bowled tightly enough on a lifeless surface. Seamer James Anderson had taken 1-67 and Panesar 1-69.
The opening partnership was the 10th-highest in New Zealand Test history before Fulton snicked an Anderson delivery to tumbling wicketkeeper Matt Prior.
It was the lanky opener's second Test half-century.
He contributed towards the best New Zealand opening stand since Stephen Fleming and Mark Richardson put on 163 in the third Test against England at Nottingham in 2004.
It was also the second-highest partnership for any New Zealand pair in a Dunedin Test, pipped by the 164 which Daniel Vettori and Brendon McCullum put on for the seventh wicket against Pakistan in 2009.