It was the first game of the new A-League season for both clubs and Phoenix made the early breakthrough when Stijn Huysegems scored from close range in the 22nd minute.
The visitors levelled through Besart Berisha two minutes after the break and, two minutes into stoppage time, Franjic plundered a winner for the Roar after Thomas Broich's effort rebounded off a post.
Jeronimo Neumann scored twice as nine-man Adelaide United started their campaign with a 3-1 home win over Perth Glory.
The Argentinian saw his ninth-minute opener quickly cancelled out by Ryo Nagai but restored Adelaide's lead with a goal in the 25th minute.
Glory pushed hard for another equaliser in a bad-tempered second half, in which both Isaias and Nigel Boogard were dismissed for the hosts, but Fabio Ferreira kept the three points on the south coast with a third goal on 64 minutes.