Adil Rashid hit 69 in new-look England's total of 408 for nine, which included 14 sixes, and the leg-spinner claimed four wickets as New Zealand were bowled out for 198 to secure England's highest victory by runs in ODIs.
Buttler walloped a blistering 129, his hundred coming off 66 balls, the second fastest ever for England and five deliveries more than the 61 he took to make a ton against Sri Lanka last year.
Buttler shared a world record seventh-wicket ODI stand of 177 with Rashid as England, who dropped stalwarts Jimmy Anderson, Stuart Broad and Ian Bell for the series, put memories of their dismal World Cup this year firmly behind them.
The omens had not looked good for the hosts when opener Jason Roy fell to the first ball of the day.
But an aggressive Root, first with Alex Hales (20) and then Eoin Morgan (50) led a recovery, taking the attack to New Zealand with the ball flying to all parts of Edgbaston.
Root's hundred came off 71 balls before he fell for 104, one of four wickets for Trent Boult.
Buttler and Rashid then took over, their partnership surpassing the previous best seventh-wicket ODI stand of 130 by Zimbabwe's Andy Flower and Heath Streak against England in 2001.
New Zealand, who thrashed England in February on the way to finishing as World Cup runners-up, made a stuttering start to their reply when Steven Finn bowled Brendon McCullum (10) and the paceman also accounted for Martin Guptill (22).
Kane Williamson (45) and Ross Taylor (57), who needed treatment after Morgan hit him on the hand with a shy at the stumps, tried to carry the fight to England.
But needing to keep up with a growing run-rate, New Zealand lost Mitchell Santner and Luke Ronchi to successive deliveries from Rashid and any lingering hopes of an unlikely victory ended in the next over when Finn trapped Taylor lbw.
Rashid completed a memorable match with a googly which flummoxed Matt Henry to return figures of four of 55 and Finn claimed four for 35.
England's previous best one-day international total was 391 for four against Bangladesh in Nottingham in 2005.
The next game in the five-match series is at The Oval on Friday.
(Reporting by Justin Palmer; editing by Ed Osmond)
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