Dylan Sage's try put South Africa on the board one minute into the second spell, but Collier's second two minutes later restored the Kiwi lead.
Kyle Brown added a second for the Blitzboks, with New Zealand having the final say through Luke Masirewa.
Skipper Scott Curry said the All Blacks Sevens had known how crucial a strong start was against the series-leading South Africans.
"We knew they were going to be physical and they really were at the start. We wanted to take it to them and I think we did pretty well," he said.
"We had three pretty tough games today so to get through it with three wins in a pool like that, we're really pleased."
New Zealand had earlier beaten Scotland and Russia in contrasting matches, battling to defeat the Scots 31-26 before dissecting Russia 31-5.
The All Blacks Sevens are currently second in the standings, 10 points behind South Africa, heading into the sixth round of the series.