It was Scotland's first win in Australia for 30 years since their 12-7 victory at Brisbane's Ballymore stadium in 1982.
The Scots humbled the Wallabies for the second time in three years with a gutsy defensive display in the second half to pull off a win in atrocious wet and windy conditions.
Scotland last rocked the Wallabies in their 9-8 upset at Murrayfield in November 2009 to end a 16-game losing run against Australia.
Laidlaw kicked the Scots to victory in Australia with three penalty goals from four shots while debutant centre Mike Harris landed two of his four kicks in the try-less arm wrestle.
It looked as though Scotland were going to settle for a 6-6 draw until they earned a scrum penalty in front of Australia's posts for Laidlaw to slot home to wild jubilation among the Scottish players.
Australia laid siege to the Scottish try-line in the second half but the Scots held firm with resolute defence to deny the Wallabies the game-breaking try.
It was another low point for the Wallabies, who lost to Samoa 32-23 in their first international last season.
The Scots had conducive atrocious wet conditions and ran with a stiff wind in the opening half.
The Wallabies were under early pressure and Scotland dominated possession and territory with Laidlaw putting them ahead with two penalties inside the opening 28 minutes.
But Australia's forwards began to control the ball and were rewarded after 12 phases inside Scotland's quarter to receive a penalty for off-side.
Harris, with Berrick Barnes holding the ball to prevent it from toppling off the kicking tee, landed the penalty to reduce Scotland's lead to three points after 33 minutes.
The Wallabies did well to go in at half-time only down by three after running into the teeth of a gale.
Harris levelled the scores three minutes after the interval with his second penalty, as the Australians sought to take full advantage of the following wind.
Rob Simmons, who replaced injured lock Sitaleki Timani, went over the Scottish try-line in the 56th minute but the video referee ruled replays were inconclusive and no try was awarded.
The Australian forwards kept pummelling away at the line but the Scots defence held firm against the mounting weight of possession against them.
So frustrated were the Wallabies in their efforts to break down the resolute Scotland defence that Barnes had a drop goal attempt with seven minutes left but it was wide.
South African referee
Jaco Peyper penalised Wallabies' prop James Slipper for losing the bind
in a collapsed scrum and Laidlaw kicked Scotland to a famous victory.