Linda Motlhalo gave Desiree Ellis’s team the lead on the half hour mark and Thembi Kgatlana doubled the advantage 11 minutes into the second-half.
But two strikes in five minutes from Sophia Braun and Romina Nunez brought the teams level.
“I’m very proud of how we came back. We never gave up,” said Braun. “Obviously we didn’t start the way we wanted to but we showed a lot of heart and fight to claw our way back “
South Africa, who beat Morocco to claim the 2022 Africa Cup of Nations, displayed the same nervousness in a position of strength that dogged that campaign in Morocco.
“We need to show a little bit of extra effort and take our chances,” said Ellis.
“It’s the little details. We had the chance to win this game and, again, we just gave it away. We created many opportunities. We spoke about decision-making and the importance of playing a pass to someone in a better position and, at times, we did not do that.
“When you create beautiful chances and don’t put them away, the result can end up like this.”
Kgatlana, who was awarded the player of the match accolade, added: “I think after scoring the second goal we thought we had the points and fatigue was also starting to play a role.
“Our decision-making wasn’t the best. After conceding the two goals we managed to stand our ground. It’s not a bad result but disappointing. We were up but had to accept the draw.”
Both Argentina and South Africa lost their opening games to Italy and Sweden respectively but still have a chance of advancing to the last-16 from Group G.
On 2 August, Argentina take on Sweden at the Waikato Stadium and South Africa play Italy at the Wellington Regional Stadium.
“We need to put the result against Argentina behind us,” added Kgatlana. “If we want to go through we have to give whatever it takes to win against Italy.”
At the Sydney Football Stadium, the European champions England edged past Denmark 1-0 on Friday to take a big step towards the last-16.
Lauren James spectacular strike from 20 metres after six minutes set up Sarina Wiegman’s side who struggled for fluency in their first Group D game against Haiti.
