Vernon Philander took his sixth five-wicket haul in just seven tests as the Black Caps were bowled out for 275 before Graeme Smith and Alviro Petersen eased South Africa to a comfortable advantage in the final 15 overs of the day.
Smith was on 34 at the close of play with first-innings centurion Petersen on 38 as the visitors moved to 75 without loss in their second innings to add to their tally of 474 for nine declared in their first.
Taylor was hit just above the left wrist by Morkel's delivery and, after facing one more ball, walked off the field at Basin Reserve to undergo an X-ray, which showed he had broken his left ulna.
"When your captain gets injured and has to come off it's devastating for the team and things change with that," New Zealand assistant coach Trent Woodhill told reporters.
"The team wants to make sure they can look after their skipper by putting in a performance tomorrow morning or tomorrow afternoon and make sure that we back Ross up." Taylor did not return to the crease in New Zealand's first innings and was "unlikely" to make any further appearance unless it was absolutely necessary, Woodhill added.
"If it's not going to damage his hand then he will probably bat, but that comes down to (facing) balls rather than overs and (we) will cross that bridge when we come to it.
"He's a resilient character and if he thought what was best for the team he would, but he won't be batting at (his normal position of) four tomorrow." Philander, playing just his seventh test, finished with figures of six for 81.
The 26-year-old, who took 10 wickets in the second test to help South Africa to a nine-wicket victory and 1-0 series lead, also claimed his 50th test victim when he bowled Doug Bracewell for a second-ball duck.
He has taken 21 wickets this series at 14.09.
Mark Gillespie hit two boundaries off Philander to ensure New Zealand just avoided the follow on target before he became the South African's sixth victim of the innings.
"I think we would have more than likely made them follow on, but obviously we've missed it by one run thanks to me," Philander said with a grin.
"Yeah we would have made them follow on it's just a pity we didn't get it. It's all in the start tomorrow morning. Hopefully we can get 350-plus and set them a good target."
