Resuming on 65-1 in reply to the Kiwis' 261-9, England's Moeen Ali failed to add to his overnight 23 as the county side were reduced to 143-6 at lunch.
But Ross Whiteley hit an unbeaten 103 to allow the hosts to declare on 291-7.
Martin Guptill made an unbeaten 61 as the tourists, lacking some of their Indian Premier League players, reached 89-2 at the close, leading by 59.
Whiteley, who added 80 with Ben Cox (35) and 81 with Ed Barnard (26 not out) had scored two County Championship centuries for Derbyshire in 2011, but although he reached his maiden first XI ton for Worcestershire before the declaration came, this match does not have first-class status.
This is New Zealand's final warm-up game before the first Test against England begins at Lord's on Thursday.
New Zealand's bowlers were given a decent workout by Worcestershire's batsmen on day three of the tour match.