Worcestershire made 291 for seven declared – with an unbeaten ton from Ross Whiteley – in reply to New Zealand’s 261 for nine, but Guptill rattled off 61 runs from 85 balls before the close as the tourists reached 89 for two ahead of the final day.
Worcestershire resumed on 65 for one and quickly suffered setbacks as Moeen Ali, Tom Fell and Alex Gidman departed in quick succession to make it 75 for four.
Opener Daryl Mitchell’s painfully patient innings came to an end when he was caught off Doug Bracewell for 22 off 111 balls and Worcestershire looked in trouble when Alexei Kervezee’s departure, bowled by Ben Wheeler, left them at 130 for six.
But Whiteley had other ideas and set about the New Zealand attack, hitting 18 boundaries on his way to an unbeaten 103.
The wickets had been shared between New Zealand’s bowlers, with Bracewell taking two for 72 and Neil Wagner two for 74.
Worcestershire declared with a lead of 30 after he reached three figures, but Guptill ensured that did not last long.
Even as Tom Latham (three) and Hamish Rutherford (eight) failed, Guptill hit seven fours and two sixes to quickly pile on the runs, giving New Zealand a lead of 59 going into the final day’s play.