At 349 for nine at stumps in reply to New Zealand's 268 any further runs tailenders Vernon Philander and Morne Morkel could add on Saturday would be a bonus, de Kock said.
The wicketkeeper-batsman joined Bavuma to turn the game around with a 160-run stand for the seventh wicket.
"The mindset I had going in was to somehow shift the pressure back on to them," he said.
"I didn't want to get bogged down and the only way I knew forward was to play my natural game, but obviously a little more aggressive than usual."
De Kock made 91 and Bavuma 89 and following their departure Philander and Morkel continued the stubborn resistance with an unbeaten 47 for the last wicket.
Philander was not out 36 at the close, three runs short of being the sixth South African to complete the 100 wickets and 100 runs double, while Morkel was on 31.
"The plan is to keep going at them to try and get as many runs as we can. A lead of 80 is very handy and any more runs we get from now is a bonus," de Kock added.
Three overs before lunch, it was New Zealand with their tails in the air having taken their fourth wicket of the morning, but it was to be another 39 overs before they would strike again when Jimmy Neesham removed de Kock after tea.
It ended a quality counter-attack which saw the South African pair grow in boldness as the New Zealand seamers slipped in line and length.
Neil Wagner, who took the wickets of Bavuma, described the day as "a see-saw" and said despite conceding the lead the conditions meant New Zealand were still in the game.
"After losing the toss and being put in we're pretty happy with the position we're in. There's lot of cricket left to play," he said.
"It's obviously a lot easier now to bat on. It's flattened out quite a bit, obviously, that's quite good for us. Once the moisture in the morning goes away, after lunch it's a lot easier to bat on."
The first session belonged to the New Zealand seamers as Colin de Grandhomme removed Hashim Amla for 21 then Faf du Plessis for 22.
De Kock survived an early anxious moment against his nemesis Jeetan Patel, who has taken his wicket cheaply the four times they have met in the past month.
As his timing and confidence grew, de Kock took a four and six off successive Southee deliveries and handed out the same punishment to Wagner as he raced to his eighth Test 50 in 55 balls.
Bavuma had a let-off on five when he skied a Wagner bouncer and de Grandhomme slipped when moving to take what should have been a regulation catch.
However there was little risk-taking after that in his 160-ball innings, until he eventually fell when he hooked a Wagner bouncer to Neesham at square leg.
De Grandhomme and Wagner each have three wickets while Southee has two and Neesham one.
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