The incidents in Tripoli heightened tensions in the restive northern city, the frequent scene of deadly unrest linked to the war in neighbouring Syria.
Sunni cleric Hussam al-Sabbagh, wanted for "terrorist activity", was stopped at a checkpoint by soldiers acting on "several arrest warrants", the army said.
Angry supporters of Sabbagh, who was arrested along with a bodyguard, later exchanged fire with the army in the Sunni-majority district of Bab al-Tebbaheh, a security official said.
The security official said he stands accused of "setting up an armed band and training terrorists", and of links to Al-Qaeda and radical group Fatah al-Islam.
On Friday, a court sentenced to death two Palestinians for membership of Fatah al-Islam, which fought a deadly 2007 battle with the army in a refugee camp near Tripoli.
The Nahr al-Bared fighting killed 400 people, including 168 soldiers.
Hours after Sabbagh's arrest, police raided the Tripoli home of a suspect wanted in connection with a Beirut hotel bombing last month.
Police suspected Hassan of having provided explosive belts to two Saudi suicide bombers who blew themselves up when police raided their Beirut hotel on June 25, the source said.
Hassan was holed up in a Tripoli apartment and resisted arrest, lobbing grenades at police and exchanging fire with them for around four hours.
"As security forces tried to enter the apartment, he was killed by one of the grenades he was handling," the source said, adding that two policemen were wounded.
Hassan, 24, flew from Sweden to Lebanon seven months ago after two of his brothers were killed in Syria fighting alongside anti-Assad rebels.
