The case will now go to the Supreme Court of Appeal, which will review the murder trial.
The double-amputee Olympian fatally shot girlfriend Reeva Steenkamp through the closed door of a toilet cubicle on Valentine’s Day, last year.
Judge Thokozile Masipa, who convicted him 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.
Pistorius’s family said: “We note the finding of the court and abide by the ruling.”
He 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 by then.
Prosecution spokesman Nathi Mncube said he hopes the appeal will be “expedited” but acknowledged that the process can take a long time. “We’re happy,” he said.
The approval of an appeal represents a victory for South African prosecutors, who had been disappointed that Pistorius was acquitted of murder.
The appeals court has panels of three or five judges and does not meet again until February 15, according to the website of the court, which is based in the city of Bloemfontein.
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 anyway.
While approving an appeal against her own verdict, Judge Masipa rejected the prosecution’s argument for an appeal against the sentence for culpable homicide. Prosecutors had said it was too lenient.