Klitschko, a Ukrainian former world champion boxer, has had full custody of the girl, Kaya, for most of her life, and she has lived with him in Ukraine. Kaya is Panettiere's sole heir.
The filing in a Los Angeles County court cites items seized in a federal investigation of the actor's death, unnamed and unauthorised people accessing her residence in Los Angeles, and valuables that are missing.
The petition says it is "urgent and essential" Klitschko be granted the legal status of temporary guardian of his daughter's estate.
"Without an immediate appointment for these purposes, there is a high probability that property belonging to Hayden's estate may be lost, damaged or dissipated before a duly appointed personal representative can be in place to protect it," the filing says.
Klitschko's lawyers filed the petition on Thursday, but it did not become publicly known until Monday, when it was reported by TMZ.
The filing emphasises that Klitschko's intention is solely to protect his daughter's future assets until a permanent administrator can be put in place.
Panettiere, the former star of the TV series Heroes and Nashville, died on August 16 at age 36.
A coroner's report obtained by The Associated Press said her death was from an accidental overdose of several drugs, including fentanyl, and she had been in rehab just weeks before. She was found unresponsive in an apartment in Greenville, South Carolina.
The petition says federal agents have taken Panettiere's property in an investigation, and there is no authorised person to deal with them and assure it is returned.
The filing says before her death, Panettiere and Klitschko hired security to protect her home because an unnamed person had gained unauthorised access.
The petition says it's expected the same person will continue to try to "misappropriate" property belonging to her estate.
A hearing on the issue is scheduled for October 8.
In her memoir released earlier this year, Panettiere acknowledged being addicted to alcohol and having postpartum depression after her daughter's birth in 2014.
Kaya went to live with Klitschko full-time when she was two. Panettiere's father told investigators that when she died she had plans to visit her daughter in Ukraine days later.
Klitschko, 50, an Olympic gold medallist, held several heavyweight titles between 2000 and 2015. He and Panettiere began dating in 2009 and were engaged in 2013.
