All the batsman could do was hit it off the toe of the bat to mid on where James Baker took a diving catch. Hicks had made 47.
The Otago middle order then collapsed for the second time in the match.
Nos 4 to 8 made just 20 runs between them. Wagner, using his now familiar short of a length attack effectively, took the first four Otago wickets to fall.
Baker and Colin de Grandhomme were also wicket takers.
Trent Boult couldn't claim any early scalps although came back to claim Jacob Duffy late in the order.
At 148 for 7, Otago's lead was just 120 and an early finish was on the cards.
But Rutherford, who scored his only test century in his debut match against England in 2013, found a great ally in South African Christi Viljoen who looked way more accomplished than a No 9 while he and Rutherford added 59 together.
It was Ish Sodhi who made the breakthrough, enticing the right hander into a big heave on the leg side. All he did was hit to Daniel Flynn at mid on who took a comfortable catch.
Rutherford's innings was a marked by intense concentration against a testing and quality international attack.
He faced down the attack from his old Otago teammate Wagner by regularly swaying out of the way of the bouncers, while occasionally being prepared to pull and glance on the leg side.
One such glance very nearly brought about his downfall when he glanced one to Darryl Mitchell at leg slip on 94.
The chance went begging and Rutherford brought up his century with a shot to backward square leg for 4.
It was the 190th ball he'd faced, and the 14th boundary.