Facing a series defeat after an upset loss in the first match in Hamilton, England bounced back after standout performances from Anderson and up-and-coming batsman Joe Root.
England's top order rose to the occasion, with half centuries to captain Alastair Cook (78), Jonathan Trott (65) and Root (79), the 22-year-old tipped as a future star.
But the foundation for England's victory came from a miserly five for 34 from Anderson, who swept through New Zealand's batters to restrict the home side's total.
His scalps included both the Black Caps opening batsmen and a rejuvenated Ross Taylor, whose century late in New Zealand's innings threatened to set England a formidable total, along with a quickfire 74 from 36 balls from Brendon McCullum.
The Black Caps made a slow start after being sent into bat on a tame wicket, all out for 269 after 48.5 overs, with England reaching the target of 270 after 47.4 overs.
Missing injured opener Martin Guptill, New Zealand were pedestrian, with the first boundary coming when Hamish Rutherford pulled a delivery from Anderson in the fifth over.
Anderson dismissed openers BJ Watling (7) and ODI debutant Rutherford (11) cheaply as New Zealand's frustration mounted and the home side were on just 22 runs after 12 overs.
Kane Williamson and Ross Taylor gradually increased the run rate, putting on 72 runs together to take the Black Caps to 91 before Williamson (33) dragged a ball from Chris Woakes on to his stumps to again slow the host's momentum.
New Zealand's struggle was apparent when Taylor, in his best performance since returning to the international fold after being axed as captain late last year, brought up his half century off 81 balls with only three fours.
Backed by McCullum, his successor as captain, Taylor notched a century off 116 balls with a four but was dismissed by the next delivery he faced.
The New Zealand tail failed to fire, leaving England with a tricky but achievable target.
England's Ian Bell and Alastair Cook made a brisk start to the run chase with an opening partnership of 89 in the first 20 overs before part-time spinner Williamson dismissed Bell for 44.
The England batsman attempted to cart Williamson's delivery over mid-wicket but was out when Hamish Rutherford made an athletic dash to catch him on the boundary.
Cook brought up a chanceless 50 from 67 balls, including five fours and one six, bit he was dismissed on 78 after hitting the ball straight back down the wicket to Tim Southee.
Trott and Root then combined the see England home, with Root displaying a wide variety of shots ranging from authoritative drives to risky reverse sweeps.