Cheika wants 'edgier' Wallabies
Samu Kerevi scored two of Australia's four tries, while forwards Adam Coleman and Dean Mumm also crossed the Pumas' line in the first Rugby Championship match played outside the southern hemisphere.
Bernard Foley kicked Australia's remaining 13 points on Saturday but the resilient Wallabies were a man down for 20 minutes after both Coleman and Michael Hooper were yellow-carded for high tackles.
This was just Australia's third win in nine Tests since they were beaten by New Zealand in last year's World Cup final at Twickenham. In June, the Wallabies went down 0-3 in a home Test series to an England side coached by former Australia boss Eddie Jones.
They will have a chance to turn the tables when they face England at Twickenham on December 3 as part of their end of year tour.
"I just want to have a bit more edge around everything," said Cheika when asked what he wanted Australia to improve before they returned to London.
"We need to start chasing what we want out of a game a little bit harder. The team's been changing so much, I probably didn't realise it would change as much as it has. A lot of our guys have really stood up to be counted and we're creating some competition, and that should bring a bit of edge with it."
Victory saw the Wallabies finish runners-up to the All Blacks, who won all six of their Rugby Championship matches.
Next up, Australia have the daunting challenge of trying to stop an outstanding New Zealand side from setting a new world record of 18 successive Test wins when they meet their trans-Tasman rivals at Eden Park on October 22.
What Cheika labelled a "massive task" is made all the harder by the fact that New Zealand are unbeaten at Eden Park in the professional era. The All Blacks have won a remarkable 35 successive Tests at their 'fortress' since losing to France and drawing with South Africa in Auckland back in 1994.
"I'll start thinking about that tomorrow. We've got to let the boys enjoy this victory," he added.
Whether another Rugby Championship match is staged in the northern hemisphere remains to be seen but Cheika was impressed by the size of Saturday's attendance.
"You can't say much more about 48,000 fans coming to a game of two neutrals," he stated.