The 25-year-old Aussie was brought to Elland Road by former boss David O'Leary but failed to break into the first team. Milosevic will be back-up to Scotland international Rab Douglas.
He said: "This is a great opportunity, it would have been crazy not to have come up here and give it a shot. But it was definitely out of the blue.
"I knew they had a lot of great goalkeepers but I will give it a shot. I am positive about it. I would not come here if I didn't think I could do a job. It's a chance of a lifetime for any player."
Celtic boss Martin O'Neill moved quickly to fill the gap created by Magnus Hedman moving to Italian club Ancona on loan.
"He's cheerful, wants to play and wants to have a go at it," said O'Neill.
He was frozen out at Leeds by the impressive form of Scott Carson, who leapfrogged him into the reserves after the departure of Nigel Martyn to Everton.
Milosevic will be included in the squad to face Kilmarnock in the Scottish Premier League tomorrow.