The visitors were reduced to 90 for four in their second innings by the close of play on Monday, chasing an improbable 481 to win the match and face the task of batting all day on the fifth and final day to salvage a face-saving draw.
New Zealand, however, had importantly already dismissed Jonathan Trott and England captain Alastair Cook, the players seen as most likely to withstand their attack for another 90 overs on the Eden Park pitch.
While Gooch agreed New Zealand were favourites to wrap up victory, the former England captain felt there was enough resolve still within the team to salvage a draw.
"Our backs are to the wall but there is no way in our dressing room are we going to give up, that's what test cricket is about, that's why it's called test cricket," Gooch told reporters.
"Sometimes you're put under pressure. We haven't played the cricket we would have liked in this game, but there is still a chance to salvage something on the last day.
"We have some good players in our dressing room, some good young players and we have to believe we can save the game."
England had entered the series as the world's second best test side and were expected to dominate a New Zealand team fresh off a series loss in South Africa and an administration being pilloried for its handling of the sacking of former captain Ross Taylor.
Cook's team were also gearing up for a mammoth of year of test cricket, facing back-to-back Ashes series and there was a suggestion by both local and British media their focus was a little further down the track than the three-test series in New Zealand.
Gooch, however, dismissed any notions of the team's focus being anywhere other than in New Zealand, and it would now need to be laser-sharp on the 20.12 metres long Eden Park wicket on Tuesday.
"I think we only want to play out the three sessions tomorrow. We can't look forward to the summer now," he said.
"What we have to do is make sure that every player that has got something to contribute tomorrow, his mind is right on the job ... and if the ball is there to score off you score off it, and if it's not, you don't.
"I know that's simplifying it but that's what they need to do. You fight and compete every ball, that's all you can ask."
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