Erkki Kourula, an ICC judge, said the pre-trial chamber's decision to not include genocide was based on an "erroneous standard of proof".
The court's pre-trial judges will now have to rule again on whether to add genocide to list of charges against al-Bashir.
The Justice and Equality Movement (JEM), Darfur's main armed, anti-government group, welcomed Wednesday's ruling.
"This is a correct decision," Ahmed Tugud, JEM's chief negotiator said. "We believe that what we have seen on the ground in Darfur amounts to a crime of genocide."
But Khartoum said the ruling served only to hinder the country's elections, due in April.
"This procedure of the ICC is only to stop the efforts of the Sudanese government towards elections and a peaceful exchange of power," Rabie Abdelati, a senior information ministry official, was quoted by the Reuters news agency as saying.
The decision came as al-Bashir visited Doha to meet Qatari Emir Hamad bin Khalifa al-Thani for talks on the Darfur peace process.
Sudan's foreign ministry, which described the ruling as a "political act", said the ICC aimed to "jeopardise the current peace process in Doha".
The Netherlands-based ICC indicted al-Bashir on seven charges of war crimes and crimes against humanity last March, but said there was not enough evidence to charge him with three counts of genocide.
Moreno-Ocampo appealed that decision, arguing that prosecuting al-Bashir for genocide did not depend exclusively on whether it could be proved that the Sudanese head of state had genocidal intentions.
Sudan's government has persistently denied the accusations, and al-Bashir has said the warrant against him is "all lies".
He also says the ICC has little power to enforce the arrest warrant.
Tania Page, reporting for Al Jazeera from The Hague, said that Moreno-Ocampo's apparent victory at the appeals court did not mean an arrest was any closer.
"The ICC doesn't have its own police force, so even though the issue for arrest on war crimes and crimes against humanity against Omar al-Bashir has been in existence for over a year now, he hasn't been arrested.
"He's travelled without impunity to half a dozen countries, throughout Africa. He's lived his life as normal, so to speak."
The Arab League last year issued a statement rejecting al-Bashir's indictment, while the African Union has said it will no longer co-operate with the ICC over the arrest warrant.
Hassan Meki, the chancellor of the International African University in Khartoum, Sudan's capital, says the ICC currently finds itself in a very weak position.
"The African Union is behind Bashir, the Arab [League] is behind Bashir, and countries of the Non-Aligned [Movement] are behind Bashir," he told Al Jazeera.
"They think this is a vague and a false accusation against a legitimage president, and they think that at this time, something like that will disturb the process of democratisation [in Sudan]. And Africa needs peace - not more problems, not more wars, not more tests."
According to UN estimates, around 35,000 people have been killed in Sudan's Darfur region since 2003, a further 300,000 have died from hunger and disease, and some 2.7 million were displaced.
However, the Sudanese government puts the number of people killed in the violence around 10,000.
Al-Bashir is the first acting head of state to be indicted by the ICC, and is the most senior figure pursued by the court in The Hague since its inception in 2002.
The United Nations has announced that the number of Sudanese in need of food aid has spiked this year, as compared to last year number's of 4.3 million needy.
Around 11 million people are now reportedly in need of food aid across Sudan, with around half of all South Sudanese going hungry.
The UN has cited drought and tribal conflicts as causing the crisis.
The UN's World Food Programme is also facing a total deficit of $485m needed to fund food aid this year.