Queensland Reds player Harris was on the field for less than 10 minutes when he was called on to kick the winning penalty after replacement Richard Hibbard had entered a driving maul from the side.
Wales out-scored Australia by two tries to one but the boot of Berrick Barnes, who scored 17 points, and then Harris proved the difference as Australia clinched the three-Test series 2-0.
George North and Jonathan Davies touched down for Wales with Leigh Halfpenny adding two conversions and three penalties, while Rob Horne was the man to cross the line for the Wallabies.
If Wales thought they came close to securing their first victory Down Under in the first Test in Brisbane last week, when they trailed by just one point before being beaten 27-19, their performance in Melbourne was even more impressive.
The visitors and Six Nations champions exploded out of the blocks with a try inside the first five minutes to signal their intent.
Working the ball through eight phases from kick-off it was winger George North who crossed over the line, diving in between the tackles of Wycliff Palu and Rob Simmons to touch down just to the right of the posts.
Halfpenny, maintaining a 100 per cent kicking success rate on this tour, quickly added the extras as Wales got off to the perfect start.
Australia enjoyed the overwhelming majority of possession during the opening half but, just as in Brisbane a week ago, Wales’s defence was immense and the Wallabies struggled to make turn that possession into points.
Barnes was forced to wait until the 21st minute before his first kick at goal before adding a second on the 30 minute mark as the Wallabies ended the first half the stronger of the two teams.
Indeed, Australia snuck in front just before the interval: Barnes - who flew to Sydney and back a day prior to be present at the birth of his first child - threw a fantastic dummy before cutting a swathe through the Welsh defence on the angle.
Barnes found Horne on the outside of him for an unchallenged run in before adding the conversion to see Australia lead by a point heading back down the tunnel.
Again Wales capitalised on a slow start from the Australians, adding their second try within five minutes of the re-start.
A mistake from the usually flawless Genia saw Ashley Beck punt the loose ball upfield before setting off in pursuit with Davies following shortly behind.
Beck, making his first start for Wales, was bundled off the ball but Davies got the bounce and dived on to the ball as it skidded over the line.
The second half was a tense affair with Halfpenny and Barnes trading penalties three times as the clock ticked down to the final minutes.
Halfpenny was lucky to escape serious injury when Cooper Vuna dumped the Welsh full-back on his head after a shocking tackle.
Vuna was fortunate to escape with just 10 minutes in the sin-bin; the argument could have been made for the rugby league convert to have been shown a straight red.
Despite the extra man, Wales only managed to add six points during Vuna’s absence, conceding three from the boot of Barnes, and failing to put the result beyond doubt.
Barnes picked up an injury on the re-start after Wales penalty but remained on the field, despite looking in some discomfort, until he missed his only kick of the day inside the final 10 minutes.
The opportunity looked to be gone for Australia as Barnes pulled his penalty kick wide in the 73rd minute and Wales withstood a late charge.
But a solid line-out from the Wallabies with just second remaining on the clock, coupled by a moment of madness from Hibbard, gave sub Harris the kick that snatched the win for Australia.
The final Test is in Sydney on June 23.