Two people were killed by the two blasts near the former military academy for women and the interior ministry.
An official of Libya's top security body said the network had been linked to the bombs.
It was the first deadly bomb attack since Gaddafi's overthrow last year.
The attack happened on the eve of the anniversary of the fall of Tripoli to rebel fighters last year.
The bombs struck at dawn, one of them close to the interior ministry's administrative offices, and the other near the military academy on Omar al-Mokhtar Avenue.
The city's head of security, Col Mahmoud Sherif, said the blast outside the military academy left two people dead and four or five injured.
No casualties were reported from the other explosion, he said.
Mr Sherif blamed Gaddafi supporters for the attacks, who he alleged were receiving financial backing from contacts based in neighbouring countries.
Another official, from the Supreme Security Committee that has been supervising security matters since Gaddafi's fall, told Reuters news agency later that connections between the group and the attacks had been established.
Challenge of violence
The attacks took place as crowds prepared for mass morning prayers to mark Eid al-Fitr, the Muslim celebration at the end of the fasting month Ramadan.
Earlier this month, Libya's interim National Transitional Council handed power to a newly elected assembly, in the first peaceful transition in the country's modern history.
But violence remains a challenge for the government, with several attacks taking place in the eastern city of Benghazi in recent months.
The BBC's Rana Jawad, in Tripoli, says that the government has often blamed these attacks on Gaddafi loyalists.
For many Libyans, she says, it is easier and more plausible to believe that loyalists of the former regime are behind them, but this is difficult to assess.
Security forces have also struggled to assert control over armed men who took part in last year's uprising and who refuse to lay down their weapons.
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