Joe Schmidt's men were beaten 18-9 in last weekend's opening clash but, with Johnny Sexton back in the side as one of eight changes, Ireland controlled proceedings for long spells at AAMI Park.
The Wallabies outscored their visitors by three tries to two but it was the boot of Sexton that proved the difference as the hosts' discipline let them down - the Irish number 10 kicked 16 points to put Ireland out of reach.
Australia took an early lead when Kurtley Beale found a gap to touch down under the posts inside two minutes.
But Ireland responded swiftly after the Wallabies had Marika Koroibete sent to the sin bin for a dangerous tackle, with Andrew Conway going over in the corner for the team's first try on Australian soil in a decade.
Sexton added the extras to pull Ireland level at 7-7 before kicking his side in front for the first time with a penalty after 13 minutes.
— Irish Rugby (@IrishRugby) 16 June 2018
Another three points from the fly-half's boot made it an expensive period with 14 men for the Wallabies and the Leinster man split the posts once more after 22 minutes to make it 16-7 to the tourists.
Ireland were themselves a man short soon after, when Cian Healey was yellow carded after a collapsed maul led to a penalty try for Australia, but Schmidt's side escaped from that 10 minutes with their two-point lead intact.
Keith Earls was denied a try by the TMO following a sustained spell of pressure from Ireland during the opening 15 minutes of the second half, but Tadhg Furlong claimed his first international try at full stretch to finally break the Australian resolve.
Sexton made it 26-14 from with two efforts from the tee to surpass 700 Test points and, although Taniela Tupou's late try set up an enthralling finish, the Wallabies could not complete the turnaround in the two minutes that remained.
The series will now be decided in Sydney next weekend.
Photo Getty Images. Caption: Ireland's Johnny Sexton.