Tries from prop Taniela Tupou, Filipo Daugunu and Tom Wright gave the Wallabies the edge in a scrappy contest between the world's ninth and 10th-ranked teams in Sydney and condemned Wales to an eighth straight loss.
Wales earned a penalty try in the first half and first five-eighth Ben Thomas kicked three penalties - but they rarely offered anything by way of a backline threat as the Wallabies gained a measure of revenge for their 40-6 drubbing at last year's World Cup.
"A win is really important because of the confidence you get from it," Schmidt said.
"We stuck at it. There were some nice... moments and some work-ons as well. There's a real 'sleeves up' mentality in the group and I'm really enjoying it."
A rain shower swept over the stadium during the playing of the national anthems and the early play was cagey with Wallabies flyhalf Noah Lolesio and Thomas exchanging penalties.
Lolesio added another three-pointer in the 15th minute and Australia went further ahead when prop Tupou barged his way over the line from close range for the first try of the game six minutes later.
Wales prop Gareth Thomas was sin-binned for an infraction during the lead-up to the try but Australia had the extra man for only four minutes.
The visitors rolled a maul towards the try line from an attacking lineout and Fraser McReight was adjudged to have collapsed it, the referee pointing to the posts for a penalty try before showing the yellow card to the Wallabies openside.
The rest of the half was scoreless and it was not until six minutes into the second half that Thomas's second penalty levelled up the scores at 13-13.
Australia had shown some invention in the backline and got a reward in the 52nd minute when Daugunu ran from his left wing over to the right and cut through the defence before allowing the greasy surface to take him over the try line.
Wales thought they had levelled up the scores six minutes later but the TMO spotted an obstruction as another rolling maul headed towards the try line.
Thomas cut the deficit to two points with his third penalty after 66 minutes but Wallabies fullback Wright soon produced a rare moment of quality by running 60 metres to score Australia's third try, bringing the crowd of 35,945 to their feet.
Tom Lynagh, the son of Wallabies great Michael, had come on to make his debut in place of Lolesio and converted the try to earn his first test points and extend the lead to nine points with 10 minutes remaining.
Wales kept battering away - No 8 Aaron Wainwright's effort in his 50th test was immense - but were unable to breach the Australia defensive line.
They were, though, more competitive than in their 41-13 loss to world champions South Africa at Twickenham two weeks ago and will head down to Melbourne for next week's second test feeling they have a chance of ending their losing streak.
"It was a bit frustrating," said coach Warren Gatland. "We created some chances - I'd like some clarity on that disallowed try - but we've hurt ourselves as well."
Scrappy win from Boks
In Pretoria, South Africa battled to a fortunate 27-20 victory over Ireland in the first test on Saturday, playing for only the second time since their World Cup triumph last October and struggling to shrug off the rust.
Both sides scored three tries in a tight match at Loftus Versfeld with home wingers Kurt-Lee Arendse and Cheslin Kolbe crossing for the home side, plus a penalty try awarded by English referee Luke Pearce.
Ireland fullback Jamie Osborne scored on debut with Conor Murray and Ryan Baird adding tries for the touring side in a dramatic final five minutes.
The Springboks made many mistakes in a scrappy performance and benefited from marginal refereeing calls as Six Nations champions Ireland stayed in the contest right to the end, despite conceding after three minutes.
"It's amazing, honestly. I can't explain how great it has been to just come back and be with the boys. This team is so special. What we did in 2023 was for the people so to come back and play here is really special," South Africa captain Siya Kolisi told Sky Sports.
South Africa got off to a perfect start, swinging the ball down the line from one side of the pitch to the other before Arendse side stepped inside and burst over for a simple try.
But the Six Nations champions very quickly went on the attack, making it clear the test would be no walkover and forcing furious defence from the hosts.
After an exchange of penalties that put South Africa 13-3 up, Irish fullback Osborne scored in the 35th minute after incredible leg work from winger James Lowe, who kept the ball alive as he was being tackled into touch, to reduce the halftime score to 13-8.
Lowe’s balletic ability was central to both his team’s potential and demise. He showed a clean set of heels down the wing when the ball popped out of a ruck in the 58th minute and looked to have levelled the scores, but the score was overturned after a TMO check. Pictures showed Ronan Kelleher was on the ground when he played the ball in the ruck.
It proved a let-off for South Africa, for whom the usually reliable Handre Pollard missed three penalties to stretch their narrow lead.
There was another break for the hosts when Lowe tried to keep a long kick in play, flicking the ball back infield when it might have been smarter to let it go out. It allowed poacher Kolbe to hack it forward and score in the 65th minute, with Pollard converting to stretch South Africa's lead to 12 points.
Another lengthy TMO check determined Lowe had not put a foot into touch as he retrieved the ball, showing incredible dexterity and meaning the try stood.
With five minutes left, the mistake seemed a death knell for Ireland but three more tries were exchanged in a breathtaking finish.
Veteran Ireland scrumhalf Conor Murray broke through to dive over under the posts, only for South Africa to go down the other end, win a scrum and dramatically drive the Irish back over the line for a penalty try.
Ireland refused to give up, however, and with South Africa down to 14 men after Arendse was yellow-carded, Baird went over in the corner two minutes from time to reduce the winning margin to seven points.
"It was a tough battle and we knew it was going to be very difficult with the quality of South Africa," Ireland captain Peter O'Mahony said.
"There are a lot of positives to take out of our performance but little bits here and there, playing against a side as good as this, they are going to punish you We'll go and have a look, fix a few things and look forward to next week."
The sides meet again on Saturday in Durban.
Scotland romp over Canada
Scotland scored 11 tries in a comfortable 73-12 victory over Canada on Saturday as they made a successful start to their four-match tour of the Americas.
Canada skipper Lucas Rumball went over for the first try after seven minutes in Ottawa but it was Scotland who then dominated with tries from Dylan Richardson (2), Josh Bayliss and debutant Arron Reed securing a 24-5 lead at the break.
Reed added another try in the second half along with two for new scrumhalf Gus Warr and one try each for Harry Paterson, Jamie Dobie, co-captain Stafford McDowall and Kyle Steyn.
Canada also crossed again when lock Kyle Baillie forced his way over.
Scotland meet the United States in Washington on Friday, followed by tests in Chile and Uruguay.
