Lawson was originally expected to skipper Scotland A against Tonga but was drafted into the full Scotland side when Mike Blair withdrew through injury and then took the pitch after only 21 minutes as a replacement for Chris Cusiter.
The scrum half told www.scottishrugby.org:
"The result will stand in the history books and enjoy this winning feeling and it hope it continues.
"We were all well aware of the fact that we hadn't beaten Australia since 1982 and there are players who have won Grand Slams for Scotland who have never beaten Australia.
"But in saying that we have to keep our feet grounded and not get carried away."
To read Lawson's comments in full, click HERE
Meanwhile, On Friday evening, Rory's team mate Scott Lawson started at hooker as Scotland A thumped Tonga by 38 points to 7 at Netherdale as the Scottish contingent enjoyed a good weekend.
Over in Dublin, Akapusi Qera and Apo Satala were in the Fiji back row against Ireland but were unable to have much influence as the home side cruised to a comprehensive 41-6 victory.
Finally, Eliota Fuimaono-Sapolu was a second half replacement for Samoa in Paris but finished on the losing side with France emerging 43-5 winners.