After Wright's runs and Finn's wickets at Pallekele, Stuart Broad's team are likely to need another success against hosts Sri Lanka when they return on Monday if they are to book a trip back to Colombo for the semi-finals.
Finn took three for 16 and then Wright (76) engineered a well-drilled chase - achieved with seven balls to spare - meaning England have at least given themselves a second chance after their defeat against West Indies at this same venue two days ago.
Finn perhaps completed the hardest and most telling half of the equation, as only James Franklin (50) cut any ice for the Kiwis.
But Wright made sure, with a 33-ball 50 containing three fours and three sixes.
Finn took two with the new ball, including danger man Brendon McCullum, and then returned in the 17th over to see off another of the Kiwis' best hitters Ross Taylor.
New Zealand won the toss but were soon minus opener Martin Guptill, lbw to a swinging yorker from Finn.
England's strike bowler then picked up the prize wicket of McCullum, making room to hit him over the ring to the short off-side boundary but succeeding only in edging down to Wright at third-man.
Graeme Swann, operating in his accustomed post-powerplay role, immediately had Rob Nicol mistiming a slog-sweep into the hands of deep midwicket.
By the time Kane Williamson was fourth out in the 12th over, caught-behind trying to cut the returning Danny Briggs, it was clear New Zealand were having to reassess what a par total might be.
They remained conservative until Franklin took 16 runs, including a big hit over midwicket for New Zealand's first six, from Briggs' final over.
With five to go, the total was therefore 96 for four - with Taylor and Franklin established.
Finn did for the former, who clumped a pull straight to deep midwicket, and in the same over his aggravating party piece of knocking over the stumps in delivery denied Franklin a perfectly-struck and well-deserved four past mid-off - umpire Asad Rauf signalling dead-ball.
Franklin stayed the course, though, for his 33-ball half-century as he and Nathan McCullum did enough in the final three overs to narrow England's advantage.
It was tighter still after Craig Kieswetter went for just four from 14 balls at the start of the reply, bowled when he missed a sweep at Daniel Vettori.
Alex Hales had taken 16 off Tim Southee's first over but then went on the charge to McCullum and missed an off-break, which hit leg-stump.
Wright was joined by Morgan, and the third-wicket pair barely put a foot wrong in a stand of 89 in 10 overs.
They were unfazed by the task in hand, and took the time available to them to build up the required momentum.
On a pitch favouring spin, they allowed New Zealand's frontline slow bowlers to concede only 42 runs in eight overs - but made sure they took toll of pace.
The run rate never spiralled thanks to fine placement, by Morgan in particular in his near run-a-ball 30, and at least one successful big shot in each over.
Wright drove especially well - and when the hapless Southee returned for his second attempt, England's number three hit him for one six to bring up the hundred and a second high over wide long-on to pass his own 50.
Morgan eventually fell - very well-caught by a diving Doug Bracewell off Kyle Mills at long-on - and Wright followed him, holing out to cover in the penultimate over.
But there was no longer any doubt about an outcome which ensures England will defend their only International Cricket Council title to date with at least a degree of honour.