Bab Tibbaneh is a stronghold of the Sunni factions, while the inhabitants of Jabal Mohsen are mainly supporters of the Shia Hezbollah-led opposition.
The Alawite faith is a small offshoot of Shia Islam and its adherents dominate Syria's ruling party.
Rocket-propelled grenades were being used in the renewed clashes after hours of sporadic sniper fire. One rocket hit an apartment building near the vegetable market setting it ablaze.
The latest fighting comes as the newly formed cabinet aimed to draw up a policy agenda ahead of a parliamentary vote of confidence which would enable the government to be officially installed.
The Tripoli fighting casts a shadow over the deal that ended a bitter political conflict between Lebanon's Sunni-led governing coalition and an opposition alliance led by Hezbollah, which is close to Alawite groups in Tripoli.
The rival alliances formed a national unity government on July 11 after weeks of wrangling over portfolios.
However, there is still disagreement over the government's policy statement, with Hezbollah's weapons at the heart of the dispute.
The cabinet decided on Thursday to postpone talks on Hezbollah's arms.
Hezbollah, which is backed by Syria and Iran, briefly took control of parts of Beirut in May after routing supporters of the governing coalition.