The Black Caps were struggling at 36-3 in their second innings at Friday's close, needing a further 218 runs to reach a target of 254 after England paceman Gus Atkinson struck twice and Josh Tongue dismissed key batsman Kane Williamson.
England's strong position in their first Test since a 4-1 Ashes series loss in Australia also owed much to debutant Emilio Gay.
The opening batsman top-scored with 57 in England's second-innings total of 226 - a valuable contribution in a low-scoring game where the pitch has favoured the fast bowlers throughout.
Play, in a match already interrupted by bad weather, should have resumed at 1000 GMT Saturday but persistent early morning rain meant the pitch and square still remained covered some 40 minutes later.
The delay was a boost to New Zealand's hopes of escaping with a draw after they were bowled out for just 113 in their first innings, with recalled England seamer Ollie Robinson taking his Test-best figures of 5-39.
But with the sky in northwest London brightening considerably later Saturday, the umpires announced lunch would be taken early at 11:20 GMT in the hope, provided there was no further rain, of starting at 1200 GMT in a second session now extended to more than three hours. Play did resume breifly, before rain stopped the action once more.
This is the 150th Test at Lord's, the most staged by any ground.