Four of the Boks’ six losses this season have been by five points or fewer, but the team has turned the tables in recent weeks with a 29-26 win in France and the victory in Edinburgh.
"We’re learning how to close out games and that’s why we won this game because Scotland were as good as us on the day," Erasmus told reporters.
"The maturity of this team is starting to show because we are now winning games from tight situations."
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Scotland have beaten Six Nations champions Ireland, England, France, Wales and Australia at Murrayfield in the last two years, with a narrow 22-17 defeat by New Zealand 12 months ago their only defeat in their 11 previous tests at home.
"The current world rankings doesn’t indicate how tough it is to come and play here and we could have lost the game in the dying minutes," Erasmus said.
"We’re definitely not the finished product yet, but we are slowly growing into a better team.
"To beat a team that has become a real force in world rugby, a team that is on the up, in the middle of a good home run, is satisfying."
Erasmus could be without skipper Siya Kolisi for next weekend’s season-ender against Wales after the flanker appeared to attempt a head-butt on Scotland centre Pete Horne, who was holding him on the ground as he tried to get up from a ruck.
"I heard about it just after the game, but I haven’t seen it," Erasmus said.
"I’m sure the citing commissioner will take a look and if there was anything serious and we are in the wrong, what comes our way we have to accept.
"I know him as a player and as a man so I’d be surprised (if it was malicious)."
(Reporting By Nick Said)
