Former captain Williamson hit 132 and Tom Blundell was last man out for 90 as the hosts fought back after being asked to follow on at the Basin Reserve
Williamson's tally eclipses the 7,683 runs scored by Taylor, who retired last year
It was a first Test victory for England in New Zealand since 2008 and a 10th win in 11 matches since former Black Caps skipper Brendon "Baz" McCullum and Christchurch-born Ben Stokes took over as coach and captain
The pair have now taken 1,000 Test wickets in the 133 Tests they have played together, dating back 16 years.
The tourists raced to their total in just 58.2 overs -- boosted by blistering half-centuries to Ben Duckett and Harry Brook -- as England opened the two-Test series with the sort of attacking approach, dubbed "Bazball", that has carried them to nine wins from their past 10 Tests under coach Brendon McCullum.
England are now playing knockout cricket after a weather-disrupted campaign, declared assistant coach Paul Collingwood on Monday, and are targeting four wins to lift the T20 World Cup, starting with New Zealand in Brisbane.