"It was a heavily-loaded suicide car bomb. So far, we have counted 29 civilians dead and seven others injured," Major Abdullahi Ali, a police officer responding to the attack, told Reuters.
"We assisted 48 wounded people and carried 16 others who were killed in the blast," Abdikadir Abdirahman Adem, director of Mogadishu's AMIN ambulance service, told AFP news agency.
"Some people are missing and so the death toll may rise," Major Ali said, adding the blast had destroyed nearby shops.
AFP also reports that the al-Qaeda aligned al-Shabaab militant group claimed responsibility in a statement distributed on its Telegram messaging account. The militant group has been carrying out a series of deadly attacks in Somalia in recent months.
It said the target was a military base close to the port and claimed to have killed "nearly 30". The group frequently exaggerates the number killed in its attacks.
Mohamed Hussein, a port employee, said ngunfire had followed the blast.
Al-Shabaab is fighting to overthrow the internationally-backed government of Somalia and regularly stages deadly attacks on government, military and civilian targets in the capital and elsewhere in the war-torn country.
A port official, who did not want to be named, said activity at the port has been halted, while another employee, who also did not want to be named, said workers were sent home after the attack.
Al-Shabaab's insurgency aims to drive out African Union peacekeepers, topple Somalia's western-backed government and impose its strict version of Islam on the Horn of Africa state.
The attack comes as Somalia is in the process of electing a new government with the much-delayed presidential vote due on December 28.