The hosts were 35 for three in their second innings at stumps, 274 runs ahead after they chose not to enforce the follow on when they had bowled England out for 204 early in the final session.
James Anderson and Stuart Broad then ripped the top of New Zealand's batting order to give Alastair Cook's side a sniff of an unlikely comeback.
First innings centurion Peter Fulton (14 not out) and Dean Brownlie (13 not out), however, blunted the England attack, steadied the innings then shut up shop for the final half hour as they realised starting again on Monday was far more important than scoring runs late on Sunday.
New Zealand will probably need to bat at least a session and half, if not two, on Monday to get as close as possible to 200 in their second innings and give England a total in excess of 400 to achieve an unlikely victory in about 120 overs.
The highest winning score achieved in a fourth innings at Eden Park is 348 for five by West Indies in 1969.
No other side has scored more than 300 to win in Auckland and with the pitch now becoming a bit "up and down", according to New Zealand's Trent Boult, the hosts are favourites to achieve their first series win over England since 1999.
"There's still a lot of cricket to play. I know it feels like the world is over but it's not really," Prior, who top scored with 73 in England's innings, told reporters. "We've had a couple of bad days.
"We're now in a position where we need a bit of luck and a lot of skill, but if we come in tomorrow morning and get early wickets, who knows, it could still be a good day.
"We've got some very good cricketers who can put in some match-winning or match-saving performances.
"We're going to have to bat a lot better but we can chase these runs. Guys just have to stick their hands up.
"The thing I love about this team is that we might do it the hard way but we fight."
Anderson and Broad had looked like they were spoiling for a fight late on Sunday as they steamed in to reduce New Zealand to 8-3, but when Fulton and Brownlie quietly began to see off the onslaught, their nostrils become a little less flared and eyes a little less bulging and wide.
"I don't think 30 for three was in the plan but I think to have a 280-run lead is a pretty good position to be in," Boult, who took a career-best six for 68, told reporters.
"We just have to look towards tomorrow morning as the first hour being pretty important. We need to put a couple of good partnerships together and see where that gets us.
"I'm not sure on a figure but we will see how it unfolds."