According to Syria's state-run news agency Sana, the plane that went down had suffered a fault with its "control mechanisms" during a routine training mission, forcing the pilot to abandon the aircraft.
The aircraft was shot down near the town of al-Muhassan, around 120km (75 miles) from the Iraqi border in Deir al-Zour province, the rebels say.
A group calling itself the "Revolutionary Youth of the Land of the Euphrates" uploaded a video to YouTube purporting to show the captured pilot surrounded by three armed rebels, saying that his mission was to "bomb the town of al-Muhassan".
In the video, which cannot be independently verified, the seemingly middle-aged man identifies himself as a pilot, Col Fareer Mohammad Suleiman. He appears to have minor bruising to his face which he attributes to the plane crash.
In other footage provided by the rebels, what appears to be a Russian-built MiG-23 fighter jet is shown carrying two under-wing weapons pods thought to be loaded with air-to-ground missiles.
Anti-aircraft fire can be heard before the jet bursts into flames.
Rebel gunners are then heard on the footage celebrating.
The BBC's Kevin Connolly, in neighbouring Lebanon, says that if the rebels have succeeded in downing a MiG-23, it would be a significant moment in the conflict.
Reports have emerged recently of anti-aircraft weapons reaching the rebels, and warplanes have been seen in recent weeks strafing and bombing targets in Aleppo.
Earlier this week, the rebels posted photos online showing them with a full surface-to-air missile system. This represents a potential threat to the regime's air power, correspondents say.
Jonathan Marcus BBC Defence Correspondent
The aircraft shown in the video is a MiG-23 ground attack aircraft, a type first delivered to Syria by the Soviet Union in 1973. In the current fighting, the jets are reported to have been used in July in the bombing of targets in and around Aleppo. It is not clear from the video exactly what brought the aircraft down.
However, if it was lost due to hostile action, this would be a first, indicating that the rebels do have a basic anti-aircraft capability. The government's air power has not been a decisive factor in this conflict so far. But the loss of the aircraft gives the rebels an important propaganda victory.