New Zealand, set 254 to win, were bowled out for 138 before lunch on the fourth day, with paceman Gus Atkinson taking 5-30 as England went 1-0 up in a three-match series.
Victory in the 150th Test staged at Lord's - the most at any ground - gave England a 1-0 lead in a three-match series.
Bowlers on both sides were on top throughout under overcast conditions and on a pitch of uncertain bounce, with Ollie Robinson marking his return to England duty for the first time in over two years with a Test-best 5-39 in New Zealand's meagre first-innings 113 all out.
Emilio Gay scored 57 on England debut during a second-innings total of 226 as the hosts enjoyed a comprehensive win in their first Test since a 4-1 Ashes loss in Australia completed in January.
This series continues across London at the Oval, where the second Test starts on June 17.