Investigators brought in a navy ship for a renewed search effort on Jan 8 to find the cockpit voice recorder after the plane crashed into the Java Sea in October.
Ridwan Djamaluddin, a deputy maritime minister, said the agency investigating the crash that killed 189 people, the National Transportation Safety Committee (NTSC), had informed the ministry about the discovery.
The flight data recorder gathers information about the speed, altitude and direction of the plane with enough storage for 25 hours of data, while the cockpit voice recorder keeps track of pilots' conversations and other sounds in the cockpit.
Investigators say the plane had encountered technical problems.
Mr Nugroho said that a weak signal from the recorder had been detected "for several days".
The main body of the aircraft has never been found.
Officials, however, warned it could take up to six months to analyse data from the black boxes.
Lion Air is one of Indonesia's youngest airlines but has grown rapidly, flying to dozens of domestic and worldwide destinations.
It has now become Southeast Asia's biggest airline group by fleet.
The device was transported to a port in Jakarta, where it will be handed over to the transportation safety committee, which is overseeing the accident investigation.
The crash was the world's first of a Boeing 737 MAX jet and the deadliest of 2018.
An AOA sensor provides data about the angle at which air is passing over the wings and tells pilots how much lift a plane is getting.
Black box data showed the plane also had an airspeed indicator issue on multiple earlier flights.
The preliminary crash report stopped short of making any recommendations to Boeing but it has come under fire for possible glitches on the 737 MAX, which entered service in 2017.