Describing the first attack, Malik Mohammed Iqbal, the city police chief, said an explosives-packed car was driven into a parking lot and detonated next to the building.
He said the blast tore down the façade of the building, knocking out the walls of several offices and damaging scores of homes in the neighbourhood.
The agency mainly deals with immigration and people smuggling but the building also housed the offices of a special US-trained unit created to counter terrorism, which was possibly the main target, security officials said.
"I have never seen such a deadly suicide attack," FIA chief Tariq Pervaz told reporters outside the agency's badly damaged headquarters.
Kamal Hyder, Al Jazeera's Pakistan correspondent, said:
"Big questions are being asked, particularly because the federal investigation agency has made key arrests in the case of the attack on the High Court and again on the Naval War College, both very high and sensitive security areas.
"Now it appears according to reports that we are getting is that the intended target was the FIA because the agency had the custody of key people who were arrested. The intended target was the FIA that has emerged quite clear now."
The second explosion, also caused by a suicide car bomb, shattered the office of an advertising agency in a residential neighbourhood.
"An explosives-laden vehicle was rammed into the office," Brigadier Javed Cheema, the interior ministry spokesman said.
It was not immediately clear why the advertising firm was targeted but the office is close to the Lahore home of Asif Ali Zardari, the widower of slain opposition leader Benazir Bhutto.
The two attacks, which deepened the security crisis facing Pakistan's incoming civilian government, left rescue workers scrambling through piles of rubble and charred flesh in a bid to find survivors.
The explosions came a week after two suicide bombers struck a naval college in Lahore, killing at least five people, in the third attack to hit the previously peaceful city this year.
Hyder said the attack points to a total failure of the security apparatus.
"Earlier, the violence was limited to north west frontier province, but now the city of Lahore, which is the heart of Punjab, is attacked, which shows that whoever is carrying out these attacks is targeting the state and the attacks are well coordinated and pretty widespread." he said.