Second-rower Hugh Sinclair put the visitors in front after 12 minutes at Churchill Park on Saturday when he powered through Vinaya Habosi's tackle, but Setareki Tamanivalu reduced the gap as the former All Blacks centre burst through the defence to go over the line.
Four minutes later, however, winger Dylan Pietsch struck at the other end, keeping hold of the ball behind his back to score.
Fiji's Levani Botia dived over the line to reduce the deficit to two points but Teti Tela missed the conversion to trail 15-13 at the interval.
Filipo Daugunu's try seven minutes into the second half extended Australia's advantage before Apisalome Ratuniyarawa scored to keep the Fijians within four points.
Jeremy Williams finally gave Australia A breathing space 17 minutes from time when he charged down Ben Volavola's kick and scored a try that effectively sealed the win.
Earlier in Lautoka, hooker Ray Niuia scored a hat-trick of tries as Samoa fought back from a halftime deficit to hand Tonga a 34-18 defeat.
Niuia capitalised on Veikoso Poloniati's sin-binning early in the second half to overturn Tonga's 18-10 lead at the interval as the Samoans used the strength of their forwards to devastating effect to secure the win.
Tonga had endured a heavy defeat to Fiji in their tournament opener last week, and there was no place in the squad for former Wallabies centre Israel Folau or ex-All Blacks Malakai Fekitoa and Charles Piutau.
Despite the absence of the high-profile trio, Tonga took early control with a pair of James Faiva penalties followed by tries from Fetuli Paea and Poloniati as their side established an 18-3 lead inside 30 minutes.
But Niuia applied the finishing touch as the Samoan maul drove the Tongans back in the 35th minute and, six minutes after the break, the hooker completed a repeat performance to further close the gap.
In the 53rd minute, Niuia completed his hat-trick when he peeled off the back of the maul to score in the corner.
Tumua Manu increased Samoa's advantage when he intercepted and raced home unopposed and Theo McFarland's pick-up and drive added further to the scoreline as Samoa finally prevailed.
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