“I’m really proud,” said Australia coach Joe Schmidt.
“We’ve made things difficult for ourselves by giving them a big start at home, a big crowd and difficult conditions.
“We missed the restart and the pressure was back on us, but we fought back, scored a try and created other chances that could have been good.”
Juan Martinez Gonzalez scored Argentina’s only try early in the first half and Gloucester’s Santiago Carreras penalty gave Pumas a healthy lead going into the second half.
Jake Gordon and Rob Ballettini both scored for Australia, while Noah Loretio added a penalty and two converted goals.
The game became extremely dangerous in the final minutes and substitute fly-half Donaldson converted a simple penalty to give the Wallabies a welcome win.
Despite the win, Australia, who host the British and Irish Lions next year, remain bottom of the table behind Argentina, New Zealand and leaders South Africa.
Australia, who lost twice to the Springboks at home in August, face a rematch against Argentina in Santa Fe on September 7.
