Chasing a fifth-straight win and fourth in a row under new coach Les Kiss, the visitors benefited from back-to-back Argentina yellow cards for deliberate knock-ons that left the hosts two short for most of the final 10 minutes.
But with playmaker Gordon (leg) off injured - halfbacks Kalani Thomas and Ryan Lonergan acted as playmakers - the hosts coughed up possession and then knocked on as Los Pumas wound down the clock and even searched for their own match-winner well after the final hooter.
At least twice the Wallabies thought they'd won a penalty around the ruck but there was no whistle and eventually Argentina kicked it dead to settle for a draw.
It left coach Kiss to lament a handful of "game management" mistakes that "let them off the hook" ahead of a Perth Test against world champions South Africa later this month.
"We're not going to catastrophise it but it's one, two, three moments," he said of a contest they led 14-0.
"It's difficult to come here and do what we did, but we let them off the hook.
"It's a series win, a trophy, some good stuff in there, but we know we have things to work on."
Australia's 27-21 win in San Salvador a week earlier secured a first series win in Argentina for the side, who had copped heavy losses in the return leg on both of their previous visits.
The Wallabies led 14-0 after tries to skipper Harry Wilson and winger Max Jorgensen, after clever build-up from the forwards.
But the hosts, in four-degree weather and sleeting rain, found their feet to level with back-to-back tries.
They then threatened to go ahead after the half-time buzzer when Tom Wright gave away a needless penalty for a stray forearm after touching the ball dead for what should have been the end of play.
The Wallabies held Los Pumas out though and it was flanker Fraser McReight, hitting the ruck at pace, that gave the Wallabies the lead back after halftime.
But the hosts lifted, dominant rolling mauls and precision kicking leading to back-to-back tries that put them clear, 28-21, with 15 minutes to play.
The game turned again when Argentinian centre Lucio Cinti was yellow-carded for a deliberate knock-on with 12 minutes to play.
Seconds later fullback Santiago Carreras followed him, while also giving away a penalty try, when he stuck a hand out to deny Filipo Daugunu a certain five-pointer.
That levelled things 28-28 with 10 minutes to play, but two-try captain Pablo Matera won a crucial penalty defending the ruck and the Wallabies' efforts to find the match-winner were fruitless in a frustrating finale.
Gordon, who impressed in a first-half cameo at fullback while Len Ikitau and Daugunu were both assessed for concussions, will have scans after copping a knock to his lower leg late in the second half.
It continues a frustrating run of injuries for the flyhalf, who had pulled the strings well in back-to-back Test wins after recovering from a calf issue.
"That last 10 minutes, they played the conditions well and our pack was trying to whack them and get the ball back," Wilson, who was superb in partnership with McReight and Rob Valetini after being rotated out of the side last week.
"We put ourselves in the position to win ... we're hurting, we really wanted to win that.
"But to bring back the trophy's great .. it brings some momentum and confidence into the group."
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