New Zealand were tottering at 5-67 before Oram and Brendon McCullum combined to help them reach 210, still a disappointing total after winning the toss and choosing to bat on Tuesday.
Stand-in captain Andrew Flintoff was the pick of England's bowlers, recording 4-21 from 10 overs, while Paul Collingwood took three catches and trapped Craig McMillan lbw to put New Zealand on the ropes.
On what looked a good batting wicket, the Kiwis were soon in trouble against some disciplined England bowling, particularly by Flintoff.
Chasing their first win in the series, New Zealand lost opener Nathan Astle in the second over, caught at second slip by Collingwood off Anderson for one.
Struggling captain Stephen Fleming, who has been enduring a slump in form, fell for 20 in Flintoff's first over.
The Kiwi skipper had just gained some confidence by lofting seamer Jon Lewis for six, but got an edge through to wicketkeeper Paul Nixon from a relatively innocuous Flintoff delivery.
Fleming's dismissal sparked a collapse that saw the Kiwis slump to 5-67 in the 24th over, with Collingwood playing a central role.
He took two more catches to remove Peter Fulton (24) and Ross Taylor (15), the latter a gem as he dived to his right at first slip from the bowling of Monty Panesar.
The industrious Collingwood then capped a purple patch when he trapped the dangerous Craig McMillan, who made 89 against Australia on Sunday, lbw for just two with the fourth ball he bowled.
McCullum then joined Oram and the pair set about rebuilding the New Zealand innings.
Oram, returning from three months on the sidelines with a hamstring injury, celebrated his recall with six fours and two sixes, reaching 86 off 89 balls before being caught two overs from the end off the bowling of James Anderson.
McCullum was content to work singles and turn the strike over to his partner, while Oram made light of his recent lack of cricket with some powerful strokes as the two put on 120 for the sixth wicket.
McCullum made 30 before falling to Flintoff.
New Zealand have lost all three of their tri-series matches so far, while hosts Australia are unbeaten in four.