"A good couple of sessions on Monday and we've got a chance of forcing a win," said England's batting coach Graham Thorpe.
Vince and Stoneman came together at 24 for one after the loss of Alastair Cook for 14. Stoneman made 60 before Tim Southee dismissed him and Vince fell late in the day for 76.
"Both would have loved to have got 100s and gone on. But they will get a good pat from us in the dressing room," Thorpe added.
BJ Watling, who made 85 in New Zealand's rearguard action to lift their first innings total to 278 said the first session on Monday would dictate the outcome of the Test.
"We're probably fractionally behind. Obviously Vince and Stoneman had a really good partnership," he said. "Monday morning will be a massive session in the game. If we can take four or five wickets in that session we can put them under some pressure. But obviously if they come through it then they're in control of the game."