"I'm really proud for what we've done tonight," France captain Anthony Jelonch said. "We managed to do what we couldn't do last Wednesday."
France coach Fabien Galthie said the win by his depleted squad was a "victory for all those who support us."
He said it would kick off the Jul 14 national holiday a few hours in advance: "Let's savor together!"
The series-deciding third test will be back in Brisbane on Saturday.
France led most of the match, again. Jaminet landed two penalties and converted Damian Penaud's counterattacking try in the 21st minute to give the visitors a 13-3 lead.
The Australians had chances to take the lead, with winger Marika Koroibete crossing the line twice early in the first half but having tries disallowed both times — once for a knock on early in the movement and another for a forward pass.
The French led 16-6 with halftime approaching when scrumhalf Jake Gordon scored a roles-reversed try inside the left corner post.
The Australian forwards attacked the French line for seven phases before prop Taniela Tupou picked up from the base of a ruck and, instead of burrowing for the line, threw a cut-out pass wide to an unmarked Gordon to dive across the line.
Flyhalf Noah Lolesio, who kicked Australia's stoppage-time winner in the first test, added the sideline conversation to cut the halftime margin to three points.
He equalized at 16-all soon after the break but the French added three more penalties and led 25-16 with about 15 minutes to go.
The French knocked on over the tryline in the 55th minute but Jaminet added three points from the penalty advantage. They went close to another try on the hour, with back-rower Cameron Woki winning a lineout against the throw near his own tryline and then orchestrating an attack down the short side to get his team deep in Australian territory.
Seven phases later, the French forwards were held up over the line but Jaminet again was able to nail the penalty goal.
They were seemingly right on top on the eve of Bastille Day.
But just as in the series-opener, the Australians rallied to set up a dramatic finish.
Skipper Michael Hooper dived over in the left corner to cap a long-range try in the 71st after Lolesio threw an inside pass to Tom Banks, who cut through a big gap in the defense from behind halfway and took the ball toward the attacking quarter.
His over-the-top pass to his support hit Jaminet's hand before Andrew Kellaway regathered for Australia and unloaded to flanker Hooper to score in the corner.
Lolesio converted from the sideline to cut the gap to 25-23 and landed another penalty from 40 meters out to put Australia ahead 26-25 with five minutes remaining.
From the restart France dominated, though, eventually winning the scrum penalty that set up Jaminet to take the winning penalty and keep the series alive.
"They played a really good French game tonight, taking points, building pressure," Hooper said. "Credit to them.
But, "there's a lot of disappointing things there. We strung some nice phases together, and had them under pressure, but a really easy pressure release there for the opposition. Like I said, they were really good on our ball, but we can be much better in that zone."
