Genia scored one try and set up another in a man-of-the-match performance and was ably supported by fly-half Berrick Barnes, who kicked 12 points in a stubborn win for the Wallabies.
Scott Higginbotham and Pat McCabe also got on the scoreboard for the hosts as they narrowly held off Wales in the second half, the difference between the two sides at point being as little as one point.
Beaten 9-6 by Scotland in torrid conditions in Newcastle on Wednesday, Genia and Barnes were two of a handful of players to have played three matches in a week with the Super 15 only going on hiatus after last weekend.
But the diminutive half-back pairing were on scintillating form as they kept their side together through a second-half fightback from Wales, who were looking for their first win on Australian soil since 1969.
Leigh Halfpenny was the main threat for Wales, the full-back kicking four penalties as well as converting Alex Cuthbert’s try, but he could not quite lead his side to the win as Australia stubbornly held on.
Genia, who was apparently on the brink of hypothermia after the Scotland game in mid-week, came bursting out of the blocks and made his first break with less than five minutes on the clock before earning his side the first penalty of the night just a couple of minutes later.
Barnes, who missed just two kicks throughout the match, bouncing both of those off the right-hand post, duly got the Wallabies off the mark before the hosts blew a gilt-edged chance to score the first try on the 15 minute mark.
Having been savaged by the Australian media all week, the Wallabies clearly came out with a point to prove as they ran the Welsh defence ragged in the early stages.
But a poor decision from Rob Horne to hold on to the ball when he had Adam Ashley-Cooper free for an easy run in outside of him, saw Wales let off the hook, despite some shaky work from Priestland under the high ball.
It was not to last long, however, as relentless pressure from the Australian back line stretched Wales’s defence enough to allow Higginbotham to burrow over and open the scoring, Barnes adding an easy conversion from in front of the posts.
Genia could have doubled Australia’s lead just five minutes later when a brilliant piece of work from Tatafu Polota-Nau at the breakdown was only letdown by his kick upfield, the Waratahs hooker opting to put the ball down the line instead of kicking to the space infield. Genia gamely chased after the ball but ran out of time when it dribbled into touch.
There were nearly 25 minutes on the clock when Halfpenny finally got his side off the mark; the full-back, who landed all five of his kicks in the match, threading a tricky kick from out wide inside the left upright.
But again, it was only a temporary respite for Wales as Australia attacked anew, first rugby league convert Cooper Vuna and then Genia again fashioning breaks that were only denied but some high-quality last-gasp defending from Wales.
Barnes bounced his second penalty attempt off the upright before Digby Iaone became the latest player denied by the Welsh defence as they went in to the interval trailing 10-3, a score that could easily have been more.
The interval did not seem to deter Australia however as Genia grabbed his well-deserved try right from the re-start, single-handedly carving his way through the Welsh defence to touch down.
Barnes added another two points with the conversion before Halfpenny reduced the deficit with a second penalty.
Yet again Australia came bouncing back quickly; Genia and Barnes combining in front of the posts for the latter to kick a drop goal and re-establish the lead.
A huge hit by Polota-Nau on Scott Williams, that left both players forced off the field with injury, seemed to galvanise Wales as first Halfpenny landed another penalty before replacement Ashley Beck put Cuthbert free to run home.
A conversion and fourth penalty from Halfpenny completed a brilliant five minutes for Wales that drew them to within one point of an increasingly ragged looking Australian side and set up a barn-storming final 15 minutes as the rain began to fall in Brisbane.
But Australia, so often criticised for being able to hold on in tense situations, once again came back at Wales.
Yet again it was Genia that proved the difference for Australia, skipping through a tackle in front of the posts before feeding McCabe to cross over.
Another successful kick from Barnes was quickly followed by another break from Genia before Higginbotham produced a try-saving tackle on Cuthbert to deny Wales their last chance as Australia held on.
The second Test is in Melbourne in a week’s time before the three-Test series concludes in Sydney on June 23.