The Springboks edged a tense second test in the Rugby’s Greatest Rivalry series in Cape Town on Saturday with a 33-26 victory that leaves the four-match series level at 1-1 heading into the third test in Johannesburg next Saturday.
Rennie spoke of his unhappiness at the scrums after the game, suggesting it was "still a bit of an issue them (the Springboks) walking around," and said they would be giving feedback to World Rugby over their concerns.
But Erasmus believes Rennie is deliberately trying to paint a picture for referees of what he terms a non-issue.
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"We've been playing as a pack (of forwards) with these guys together (for some time), and it's the first time in history that people are talking about us walking around," he told reporters.
"The first six scrums for New Zealand, every single one collapsed, and it was just reset, reset, reset, reset. So why is it only on their ball that the scrum collapses?
"Don't come with that nonsense and say, ‘you're walking around’ and trying to put that out in the media.
"Their last lineout was skew. I’m not going to moan my arse off about that. So if you just want to start moaning about stuff like that, then they’re under pressure.
"Because last week we didn’t moan after there was clear head contact a lot of the time. And today in the first 40 minutes there were four head contacts, and nothing was done about it."
South Africa were dominant in the scrums in the first test, but less so in Cape Town, where New Zealand largely held their own.
"We don’t want to stay quiet and just say, ‘Oh, you’re right, we’re walking around’,” Erasmus added. “We scrum straight. We’ve never been accused of walking around. So I don’t understand.”
The third test in Johannesburg will be followed by one more in Baltimore in the United States on September 12.