Jedinak twisted his left ankle in the first half of Friday night's 4-1 tournament-opening win against Kuwait at Melbourne's AAMI Park, but played out the game.
Franjic admitted the skipper has a strong presence on the field, but whoever fills the gap will be up to the task.
"Mile Jedinak is a big loss. He's our captain and he's a great leader," he said.
"(But) there's a lot of boys in this team who can fill that role."
Oman is the No.93-ranked team (seven places higher than Australia), but Franjic insists his side won't change its tactics or its aggressive style.
"They're a good counter-attacking team," he said.
"We'll keep our up-tempo game, high pressing and really try to hit them early."
Franjic, who made his Socceroos debut in 2012, joined Russian team Torpedo Moscow last year after an extensive career with Brisbane Roar.
The 27 year-old said he'd shaken off the hamstring injury he picked up after Australia's group-stage loss to Chile in the FIFA World Cup last year, declaring playing back-to-back games would not be a worry.
"I had a good preparation pre-tournament. Obviously the Russian league finished early and I had a good pre-season so ... I'll be ready and raring to go," Franjic told Sky Sports Radio.
"We've been recovering well so there should be no excuses at all."
Australia meets Oman at Sydney's ANZ Stadium at 8pm (AEDT) on Tuesday, before facing Korea Republic in Brisbane on Saturday.
