"Many more people were wounded," the source added.
The first blast rocked the city centre near the home of outgoing Prime Minister Najib Mikati, although his office said he was not in Tripoli at the time.
The second struck near the port of the restive city with a Sunni Muslim majority, close to the home of former police chief Ashraf Rifi, a security source said.
Lebanese television channels aired footage of buildings with their fronts blown in and vehicles ablaze, as bystanders rushed to help the wounded.
The explosions come a week after a suicide car bombing killed 27 people in a Beirut stronghold of the Lebanese Shiite movement Hezbollah, which is fighting alongside Assad's forces.