Russell, who along with team-mate Allan Dell has been called up to the British and Irish Lions squad, added the extras to each touchdown with Greig Tonks’ early penalty the other Scottish score.
It was Scotland’s second successive win Down Under, having also beaten the Wallabies at Newcastle in 2012.
Israel Folau crossed over for a brace of first-half tries for the hosts, with Bernard Foley’s conversion making it 17-12 at the break as they recovered from being 10-0 down. Will Genia scored Australia’s third midway through the second half but the visitors held on for the win.
Ross Ford came off the bench for his 109th cap for Scotland to equal Chris Paterson’s appearance record.
Scotland had been the first to settle, winning a penalty inside the first minute when the hosts were spotted with hands in a ruck. Tonks stepped up and landed his first Test points from just inside the Wallabies’ half.
Both sides were happy to run the ball but neither could make much headway when they did with their defences very much on top. However, there was nothing the home side could do when a poor pass from scrum-half Genia was knocked straight into the hands of Scotland centre Taylor for a simple run-in.
Russell converted and the Scots were looking good for their lead, only to surrender the initiative when a string of penalties went against them, the final one leading to a maul on their line. The threat of a Genia break sucked in the visiting defence and a long pass to full-back Folau gave him an easy run-in and Foley landed the conversion.
Australia seemed determined to keep their opponents in the game, however, a late tackle earning Foley a spell in the sin-bin and seconds later Genia took far too long over a clearance kick to let Russell charge it down, collect the ball on the full and touch down for a try. The conversion restored Scotland’s 10-point lead.
The away side needed the gap as the Wallabies started to find space when they opened up. After new boy Eto Nabuli on the wing had come within inches of crossing in the corner, Scotland number eight Ryan Wilson was sent to the sin-bin for slowing the resulting ruck.
It did not lead directly to the try but again the resulting maul pulled in Scottish defenders. A cross kick by Foley found Folau on the touchline to leap above Tonks, claim the ball and roll over the line. Foley missed the kick, however and the gap was still five at the interval.
Soon after the break Australia were on the figh back and they did work new centre Karmichael Hunt over but he lost the ball as he went for the line and Scotland managed to clear, holding out until Wilson came back on.
Ford then came off the bench for his landmark appearance, but he could do nothing to stem the string of penalties going against his side which were giving Australia a powerful foothold in the game.
Wave after wave of attack came their way until eventually a ruck right on the Scotland line gave Genia the chance to reach out and just ground the ball on the line. With Foley converting, Australia had taken the lead for the first time as the match entered its final quarter.
Scotland quickly hit back though as they forced their way into the home 22 where Lee Jones bounced over the defence to find Taylor on a support line and put Watson in for the score before Russell converted.
Australia did their best to claw it back with Folau, Hunt and Tevita Kuridrani all within inches of scoring but somehow Scotland managed to find ways to stop them every time and they held on for the final whistle.