The 27-year-old was accused of displaying a flag in support of proscribed terrorist organisation Hezbollah at a gig in November last year, until a technical error in the way he was charged led to the chief magistrate ruling he could not try the case.
The Crown Prosecution Service said in a short statement it would be appealing against the decision because "we believe there is an important point of law which needs to be clarified".
After the hearing in which the case was dismissed, O hAnnaidh, who performs under the stage name Mo Chara, vowed the rap trio would "not be silenced".
Chief magistrate Paul Goldspring agreed with O hAnnaidh's lawyers that prosecutors needed to seek the permission of the UK attorney general to charge the rapper before informing him on May 21 that he would be charged with a terror offence.
It is understood the Crown Prosecution Service's position is that permission only needed to be obtained before his first court hearing, which took place around a month later.
The decision to appeal against the judgement comes as Attorney General Lord Hermer warned shadow justice secretary Robert Jenrick not to prejudice future proceedings against O hAnnaidh, asking him to "carefully consider any further public comments".
Jenrick had previously written to Hermer calling for an urgent inquiry into the issue, claiming it "raised serious questions about the competence of all those involved".
The chief magistrate labelled the decision to charge O hAnnaidh as "unlawful" as he dismissed the case while sitting at Woolwich Crown Court on September 26.
Prosecutors allege O hAnnaidh can be seen in a recording of a gig at the O2 Forum in Kentish Town, north London, which shows him wearing and displaying the flag of Hezbollah while saying "up Hamas, up Hezbollah".
"This entire process was never about me," he said, after the hearing last month.
"It was never about any threat to the public, it was never about terrorism - a word used by your government to discredit people you oppress. It was always about Gaza, about what happens if you dare to speak up."
In an interview with Virgin Media News, the rapper told the Prime Minister Keir Starmer "better luck next time", adding: "even if it had went to court we would have won anyway".
