De Kock went to the break at the Basin Reserve on 63, while Bavuma was 61 not out after the pair had put together a partnership of 124 to rescue the Proteas following a rocky opening session.
It was the eighth test half century for de Kock and sixth for Bavuma as they brought the tourists to within 50 runs of New Zealand's first innings 268.
After taking four wickets in the first session, New Zealand would have had high hopes of taking a significant first innings lead and then utilising the best of the batting conditions.
De Kock and Bavuma thwarted them, however, with a delightful array of cuts, pulls and drives to score 114 runs in the session at more than four an over.
The Proteas had resumed the day on 24 for two and were immediately under pressure with nightwatchman Kagiso Rabada bowled by Tim Southee for nine in the second over of the day.
Henry Nicholls, who scored his maiden test century on Thursday, then took catches at midwicket to dismiss JP Duminy for 16 and Hashim Amla for 21.
While Duminy spooned the ball to Nicholls off left-arm quick Neil Wagner, the removal of Amla was much more difficult.
The right hander flicked a Colin de Grandhomme delivery off his pads in the direction of the fielder, who parried the ball above his head then reached out to grasp it with his left hand as he tumbled to the ground.
South Africa captain Faf du Plessis had flirted with disaster throughout his innings with several lofted shots before he gave seamer de Grandhomme his third wicket, caught behind for 22 about 10 minutes before lunch.
(Reporting by Greg Stutchbury; Editing by Nick Mulvenney)
