The sensational case will go to the Supreme Court of Appeal, which will review the murder trial of Pistorius, who fatally shot girlfriend Reeva Steenkamp through the closed door of a toilet cubicle on Valentine's Day last year.
Judge Thokozile Masipa, who convicted Pistorius and sentenced him to five years in jail, acknowledged that chief prosecutor Gerrie Nel had raised legitimate "questions of law" that should be studied by the appeals court.
"This might have a practical effect" on the conviction, Masipa said.
"We note the finding of the court and abide by the ruling," Pistorius' family said in a statement.
Pistorius could face a minimum of 15 years in prison if the appeals court overturns the culpable homicide conviction and raises it to a murder conviction.
Under his current sentence, Pistorius could be released from prison and placed under house arrest after serving 10 months, or one-sixth of his sentence. It is unclear whether the appeals court will have ruled on his case within 10 months.
Nathi Mncube, the prosecution spokesman, said he hopes the appeal will be "expedited," but acknowledged that the process can take a long time.
"We're happy," Mncube told journalists.
The approval of an appeal represented a victory for South African prosecutors who had been disappointed that Pistorius was acquitted of murder.
In another disappointment for the state, British businessman Shrien Dewani left South Africa on Tuesday after being acquitted of the murder of his wife Anni during their honeymoon in Cape Town in 2010.
The appeals court has panels of three or five judges and does not meet again until Feb. 15, according to the website of the court, which is based in the South African city of Bloemfontein. Decisions are based on the opinion of a majority of judges.
"Witnesses do not appear before the court, and the parties need not be present during the hearing of an appeal. A written judgment is usually handed down shortly after the argument," the court's website says.
In arguing for an appeal, the prosecution said Judge Masipa incorrectly interpreted a legal principle. Under that principle, a person should be found guilty of murder if he foresaw the possibility of a person dying because of his actions, and went ahead with those actions anyway.
While approving an appeal against her own verdict, Masipa rejected the prosecution's argument for an appeal of the sentence for culpable homicide. Prosecutors had said it was too lenient.
In any case, if the appeals court finds Pistorius guilty of murder, the sentence would automatically be raised to match the severity of the crime.
Pistorius said he thought a dangerous intruder was in the house when he killed Steenkamp, a model and budding reality TV star; prosecutors allege he killed his girlfriend after an argument.
Pistorius' legs were amputated below the knees when he was 11 months old because he was born without fibula bones due to a congenital defect. However, he grew up playing sports with prosthetics. The sight of Pistorius racing at the London Olympics on carbon-fiber blades was one of the enduring and inspirational images of the 2012 games.
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