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A try from Ulster's Jared Payne helped Ireland lead 7-0 and they were 10-3 ahead when Stander was sent off for his challenge on Patrick Lambie.
Lwazi Mvovo's score saw the Springboks edge ahead but Munster's Conor Murray's second-half try and eight Paddy Jackson points were enough despite Pieter-Steph du Toit's try.
Best said: "We knew to come here and get a victory would be unbelievably tough and require a massive physical performance and that's what we produced.
"We dug really deep to get that win. If we had sat back and kept giving them ball we would have eventually run out of steam, so we had to take the game to them.
"There are a lot of tired boys who are running on empty now after the euphoria of the win."
Meanwhile, Ireland flanker Stander will be available despite his red card in the first half.
"We'd love to have him available, we're going to find out in the next couple of days what, if any, consequences there are from his attempted charge-down," said coach Joe Schmidt.
"So, we'd be delighted to get him back if that's possible. If we don't, this is a 32-man squad, there are 32 players working to try and be selected and it means somebody else will step up and fill the breach."
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