After bowling out New Zealand for 185, England cruised to the victory target in just 37.3 overs.
Finn and Anderson crashed through the New Zealand top order with the new ball on Saturday, reducing the home side to 11-3 after 7.2 overs. Captain Brendon McCullum hit a defiant 79 from 68 balls but New Zealand couldn't overcome its early setback and failed to post a competitive total.
Alastair Cook made 46, Eion Morgan 39 and Jonathan Trott 38 as England paced its run chase to win with plenty of overs to spare. Youngster Joe Root was 28 not out at the end, hitting the winning runs from the bowling of Kyle Miss.
New Zealand won the first match of the series by three wickets but England won the second by eight wickets and became the first team in either the Twenty20 or one-day series to post consecutive victories.
England won the previous Twenty20 series 2-1 and a three-test series between the teams begins at Dunedin on March 6.
The tourists now head with confidence into the three test series. New Zealand had some cause for concern on Saturday when McCullum suffered a side strain which prevented him keeping wicket during the England innings.
Test wicketkeeper B.J. Watling took the gloves for New Zealand while McCullum fielded in the outfield and appeared to move comfortably. The New Zealand team for the test matches is due to be named on Sunday.
Finn and Anderson will also open the bowling for England in the tests and hope to capitalize on their outstanding form during the one-day series. Their bowling was a key part of England's wins in the second and third one-dayers, containing New Zealand's scoring at the start of its innings.
Finn took 2-6 and Anderson 1-12 from their first five over spells on Saturday as New Zealand labored to 18-3 after 10 overs. It was still only 57-3 after 20 overs and 67-5 at the midpoint of its innings.
McCullum hit six fours and five sixes to substantially boost the run rate but New Zealand still feel well short of a competitive total on a ground with short boundaries.
"We felt we didn't quite in the first game," Finn said. "Maybe we were a little bit rusty.
"But playing these last two games we felt really good. We set the tone early and the batsmen have been brilliant as well."
Finn had 3-12 entering his ninth over but conceded two fours and a six to McCullum and finished with 3-27.
"It's great to be bowling at the other end to Jimmy (Anderson)," Finn said. "He's obviously one of the best, if not the best fast bowler in the world.
"So to be able to bowl at the other end and learn off him and feed off him has been brilliant."