Quinton de Kock (114) and Rassie ven der Dussen (133) both compiled centuries in the Proteas' commanding 357-4 in Pune.
New Zealand, only five days since narrowly falling short chasing 389 in a five-run defeat by Australia, collapsed to 190 all out against a powerful South Africa attack where Keshav Maharaj took 4-46 after the pacemen had done the early damage.
This was New Zealand's third loss of the tournament, but victories over both Pakistan and Sri Lanka in their last two pool games in Bengaluru should still see the 2019 losing finalists through to the knockout phase.
"It's just two wins away," said Phillips, whose defiant 60 on Wednesday came too late to turn the tide.
"So, I think if we can just keep sticking to our basics, what we do best on the field, everything will be all right."
By opting to field first, New Zealand captain Tom Latham played into South Africa's hands and the Black Caps didn't help themselves by missing a few chances, including a possible run out of De Kock.
"Obviously South Africa are incredibly strong batting first, but we've got an incredibly good bowling line-up and there was no reason that we couldn't have restricted them to a total that could have been a lot easier to chase," said Phillips.
"Playing on at the end, I definitely thought it got better as well. So, I think had things gone slightly differently, we could definitely have come a lot closer than we did today."
New Zealand suffered yet more injury woe on Wednesday, with fast bowler Matt Henry leaving the field midway through his sixth over due to a hamstring problem.
Jimmy Neesham had to bat down the order at No 9 after being struck on the wrist by the ball in the field.
Star batsman and regular New Zealand captain Kane Williamson has also missed most of this tournament with a broken thumb.
"Every team is combating injuries at the moment," said Phillips. "But we'll see what the scans and the X-rays show and we'll go from there."
A New Zealand spokesman said an X-ray had cleared Neesham of any broken bones but that a scan of Henry's injury had yet to be assessed.