Judge Thokozile Masipa announced the ruling in a Pretoria court, saying she was satisfied that chief prosecutor Gerrie Nel had raised ''questions of law'' that should be reviewed by the Supreme Court of Appeal.
''This might have a practical effect on the conviction,'' Masipa said.
Nathi Mncube, the prosecution spokesman, said he hopes the appeal will be ''expedited,'' but acknowledged that the process can take a long time.
In arguing for an appeal, the prosecution said the judge - who in October found the Paralympic champion guilty of culpable homicide and gave him a five-year prison term - 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.
''We're happy,'' Mncube told journalists after Masipa approved the appeal.
Pistorius could be released from prison into house arrest after 10 months but he now faces the prospect of a possible murder conviction. Once prosecutors appeal the conviction, the appeals court would review the case to determine if the verdict was appropriate.
Pistorius fatally shot Steenkamp, a model and budding reality TV star, in his home on Valentine's Day last year. He said he thought a dangerous intruder was in the house; prosecutors allege he killed his girlfriend after an argument.
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO_Sports and on Facebook at AhramOnlineSports.)