The National Prosecuting Authority said the sentence was "disproportionate to the crime" and could bring the justice system "into disrepute".
The Olympic athlete was jailed earlier this month for murdering his girlfriend Reeva Steenkamp in 2013. He admitted killing her but says he mistook her for an intruder, reports BBC on Thursday.
The 29-year-old was initially given a five-year term for manslaughter, but was found guilty of murder on appeal last December.
A statement released by the National Prosecuting Authority said that an application for leave to appeal against Pistorius's sentence was submitted on Thursday.
"We hope that this appeal will also clarify further the principles of sentencing, particularly in crime categories for which there are prescribed minimum sentences ordained by legislation," it said.
Prosecutors point out that Pistorius' jail term was less than half the 15 years sought by them and that he has shown inadequate remorse for the murder.
In explaining her decision to give him six years, Judge Thokozile Masipa said mitigating circumstances, such as rehabilitation and remorse, had outweighed aggravating factors, such as his failure to fire a warning shot.
She said the sentence needed to be fair to both Pistorius and the family of the deceased.
A longer sentence would not serve justice, she said, "Public opinion may be loud and persistent but it can play no role in the decision of this court."
But the sentence caused outrage among some, who argued he had been given preferential treatment because of his status and wealth.
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