Despite having a 179-run first innings lead, New Zealand lost last week's second Test at Old Trafford by six wickets with more than a day to spare.
That left them 1-0 down in the three-match series with just one to play after the drawn opener at Lord's.
But all-rounder Oram said New Zealand, who on home soil in March were beaten 2-1 by England in a Test series after going 1-0 ahead, were not that far away from their opponents.
"In a strange way Manchester has just added to our confidence because we were so close, and we know we are so close, to this England team," he told reporters at New Zealand's hotel.
"We were written off before the tour but if you take a couple of sessions out of that match, we were the dominant team.
"That is great to know but we have to finish it off with a bit more toughness, a bit more consistency.
"We just don't seem able to seal the deal and by that I mean we are not consistent enough with bat and ball.
"On a couple of occasions we've been right on top with the bat and lost a flurry of wickets and on the other side of the coin we've been bowling well - I think back to Wellington - and let them get away to a score."
New Zealand were bowled out for just 114 in the third innings of the second Test. Afterwards, captain Daniel Vettori said this was another example of how his side, be it with bat and ball, had again struggled in this crucial phase of a Test match.
And Oram added: "It is always in the back of our minds, it's a mental thing not a technical thing because we showed in the first innings (at Old Trafford) we are proficient enough to score runs.
"Maybe we get a bit too far ahead of ourselves and start thinking about totals set or totals we're chasing when we should just be thinking about the next ball."