The victory ensured England topped their Super 8 group with a perfect record, while New Zealand’s semi-final hopes were left hanging on other results.
While England has already booked their last-four berth with two wins from two matches in the Super 8 stage, New Zealand’s qualification hopes hinge on this contest.
Archer was the target of racial abuse on the final day of the first Test between England and New Zealand at the Bay Oval in November.
England will now head to South Africa on a tour which begins at the Centurion on Boxing Day.
Root seems to have silenced his critics for a while with his classy double ton in Hamilton Test.
Right-handed batsman Joe Root was declared Player of the Match while left-handed pacer Neil Wagner was declared Player of the Series.
At stumps on Day 1, Latham and Henry Nicholls were at the crease on their individual scores of 101 and 5.
Williamson, who is prepared to travel to Hamilton for Friday's second test, said he would personally apologise to Archer.
On Monday, England lost the first Test by an innings and 65 runs and conceded a 1-0 lead to the Black Caps in the two-match series.
On Friday, the visitors started the day at 241 for 4 and suffered a batting collapse after overnight batsman Ben Stokes fell for 91.
