Hugo Keenan nosed the Lions ahead with a last-gasp try in front of 90,307 at the Melbourne Cricket Ground to leave the Wallabies heartbroken and the third test in Sydney a dead rubber.
Dan Sheehan, Tom Curry, Huw Jones and Tadgh Beirne also scored tries for the Lions, who claimed their first series win since beating Robbie Deans’s Australia 2-1 in 2013.
"Andy (Farrell), he told us that we should have no doubt in our mind that we're going to win this game," Lions captain Maro Itoje, named man of the match, said of his coach.
"Let's continue to play big. Obviously, there's loads, loads of errors, loads of errors, but we played big, we got the win, and that's all that matters."
While Farrell’s team celebrated, it was a tough night for the hosts, who had taken a 23-5 lead in a belter of an opening half featuring three tries apiece.
"It hurts," said Wallabies captain Harry Wilson.
"So proud of this team. We've been written off. We came out here, we put our bodies on the line. The game was there to the very end, and to lose like that, it hurts."
With enforcers Rob Valetini and Will Skelton back from injury, Australia hit harder at the breakdown and turned possession into early points.
Wallabies fly-half Tom Lynagh booted two penalties in the opening 11 minutes before the Lions rallied.
A scrum penalty had the Lions motoring to the line where hooker Sheehan took a quick tap and flew over a pair of Wallabies for the opening try.
Australia suffered another blow when winger Harry Potter came off with a hamstring injury, bringing Tate McDermott into the game early.
However, the pressure eased with their opponents' indiscipline at the breakdown, and Australia were able to camp out at the try-line, where James Slipper burrowed over at the left corner in the 23rd minute.
Referee Andrea Piardi had had enough of the Lions' infringements and showed Tommy Freeman a yellow card, and Australia duly capitalised with a two-try burst.
Scrum-half Jake Gordon veered out of a ruck, feigned a pass and bolted five metres through a gaping hole in the Lions' line for the first.
Centre Joseph Sua'ali'i then sent home fans into delirium two minutes later as he burst through midfield and dished off to Tom Wright, who streamed over to put the Wallabies 18 points up.
Once restored to 15 men, the Lions wrested the momentum back with two tries of their own in the last minutes of the half.
Curry danced around Gordon in the corner for the Lions' second try before a Sua'ali'i ruck penalty allowed Russell to boot for the corner.
Some smart Lions passing ended with Jones carrying strongly over the line to put his team six points from the lead at the break.
Both teams struggled to impose themselves after the restart, but after Lynagh converted a penalty in the 53rd minute, the Lions' Bundee Aki cracked the game open with a line-break.
The tourists went to the left corner where James Lowe off-loaded to the barnstorming Beirne to thunder over at the hour mark.
Finn Russell's conversion made it a two-point game, and the Lions ratcheted up the pressure, with their forwards dominating the scrum.
After several close escapes, the Wallabies finally buckled in the last minute when the quick-footed Keenan stepped through a gap on the left corner.
There was an agonising pause as the Wallabies desperately appealed for high contact in a clean-out by Jac Morgan, but the try was allowed to stand, giving the Lions their first series win with a match to spare since South Africa in 1997.