The pair took the score from 52 for three in the 14th over to 222 for four in the 41st.
Leg spinner Amelia Kerr was the best of New Zealand's bowlers, taking four for 51 from nine overs, although the flight of off spinner Leigh Kasperek was promising early; she finished with two for 49 from 10 overs.
Captain Suzie Bates top scored in the chase, making 44 from 68 balls, while Katie Perkins finished 43 not out from 53 balls.
Amy Satterthwaite made 35 from 50 and took three catches as she became the ninth New Zealand woman to play 100 one-day internationals.
England left-arm orthodox bowler Alex Hartley starred with three for 44 from 9.4 overs, including the key wickets of Bates and Sophie Devine for eight.
New Zealand roared into the match when Satterthwaite caught opener Lauren Winfield for 11 at backward point off Lea Tahuhu. That was followed by Kasperek's caught and bowled of Sarah Taylor for eight and Bates' trapping opposite skipper Heather Knight leg-before wicket for eight with a ball seaming back.
Beaumont and Sciver changed the game with their counter-attacking, and the latter also benefitted from a Kasperek spill at short backward square leg on 58.
"It was a good target by England, but if one of our top order had scored big then we thought it was gettable," Bates said.
"Tammy and Nat took it away from us and although we brought it back at the end, we struggled to break the partnership.
"We've played some good cricket in this tournament, so it's not doom and gloom. It's a big game against India now."
"The partnership between Tammy and Nat was brilliant," Knight added.
"There was pressure at 52 for three. It was a tough pitch to bat on when the bowlers took the pace off, but I was pleased with the way the girls went about it.
"It's lovely the batters are standing up and putting in big performances. You want your top five getting runs and it's a credit to the hard work they've put in over the last year."
The loss ended a streak of 15 successful chases by the White Ferns.
