Persistent rain meant only around an hour of play was possible on Saturday, though it was enough for Ollie Robinson to make a decisive impact by grabbing two key wickets.
Robinson removed Rachin Ravindra with a rising delivery that kissed the off stump before trapping Daryl Mitchell in front, leaving New Zealand struggling on 55-5 heading into day four.
The tourists remain 199 runs adrift of an unlikely 254-run target heading into the final day.
Rain interruptions dominated the day, with play repeatedly delayed before being abandoned in the late afternoon.
Despite the limited action, Robinson again thrived in helpful conditions on his Test return, taking his match tally to seven wickets.
With dry weather forecast for Sunday, England are expected to complete a comfortable win and go 1-0 up in the series.
England launched their post-Ashes rebuild in emphatic fashion with a 115-run victory over New Zealand in the first Test at Lord's, wrapping up the win before lunch on the fourth day after a brutal, seam-dominated contest.
New Zealand will need another 218 runs to win the first Test against England but the hosts need seven wickets for victory after a late double-strike at the end of the second day at Lord's.
Emilio Gay is set to make his debut for England in the first Test against New Zealand, starting on Thursday at Lord's, after the hosts announced a 12-man squad for the encounter.
Kane Williamson is set to make his final appearance at Lord's as an international cricketer when New Zealand take on England in the first of three Test matches from Thursday.
