Security troops using army helicopters and police dogs launched a manhunt to find Suleiman, who officials described as "dangerous".
Prison officers suspended
Following the escape, Ziad Baroud, the interior minister, visited the prison and ordered the suspension of several police officers who were on duty while an investigation is carried out, said the state-run National News Agency.
He also ordered all officers on duty in Lebanon's prisons to be transferred to other departments within 15 days and lower-ranking policemen within two months, it said.
Fatah al-Islam became widely known in 2007, when its fighters fought the military for three months from their stronghold within the Nahr el-Bared Palestinian refugee camp in northern Lebanon.
The group is believed to include fighters from several Arab nations, including Syria.
The clashes at Nahr el-Bared left 220 fighters, 171 soldiers and 47 Palestinian civilians dead.
Suleiman was among the dozens arrested during the camp fighting and was charged with killing Lebanese troops.
He is also suspected of involvement in other bombings in Lebanon.
The Associated Press reported officials as saying that the seven other Fatah al-Islam members who tried to escape included Abu Salim Taha, who served as the group's spokesman during the fighting.
Officials also named Yasser al-Shuqairi, who is standing trial for his role in twin bus bombings that killed three passengers on a mountain road east of Beirut in February 2007.
A third man broke his back when his blanket line unravelled and he fell from a height of five metres.
Roumieh prison was built four decades ago to house 1,000 prisoners but today holds more than 3,000.
It has repeatedly been the scene of riots by prisoners demanding better conditions.