New Zealand maintained that form with the bat thanks to captain Kane Williamson (78 not out) and Jeet Raval (52) as the Kiwis reached stumps 131 runs adrift of the tourists.
New Zealand Trent Boult, centre, is congratulated by teammates after dismissing South Africa's Stephen Cook.
Play in the first Test between New Zealand and South Africa resumed on Friday after a fire alarm in the main grandstand forced the evacuation of the venue and halted play for about 30 minutes.
Their 308 was anchored by 140 from opener Dean Elgar.
Neil Wagner hit five fours and two sixes in his 32.
When New Zealand slipped to 324/9‚ Taylor resumed his innings - which was interrupted on Thursday when he retired hurt on eight with what has since been diagnosed as a torn calf.
On a day of wavering fortunes, there was nothing normal about the start of the Test where even the brown-green wicket was not the customary emerald top expected in New Zealand.
The Proteas were 1-14 in their second innings when an emergency alarm rang out around the picturesque ground, leading both teams to walk off and mingle on the side of the playing surface.
South Africa were in trouble at 22 for three when du Plessis joined Elgar to mount their rescue mission.
Williamson brought up his 16th Test century, putting him level with Taylor and one behind the New Zealand record of 17 held by the late Martin Crowe.
With South Africa 279-7, Philander added some valuable runs but he had little support from the rest of the tail as Maharaj and Kagiso Rabada were cleaned up by Boult and Patel.
BJ Watling, who contributed to New Zealand's total with 50, believed a result was still possible with "one insane session" that could change the balance.
Hello, good morning and welcome to CricketCountry's live cricket blog and updates post for the fourth day's play of the first Test between New Zealand and South Africa at the University Oval in the down south handsome city of Dunedin.