A further six people were missing and 19 suffered injuries, agencies quoted local officials as saying. Yeysk is separated from occupied Russian territory in southern Ukraine by a narrow stretch of the Sea of Azov.
Military news channel Zvezda published video appearing to show explosions aboard the Sukhoi Su-34 supersonic medium-range fighter-bomber as it plunged towards the apartments on Monday.
Russian agencies said the pilots had ejected.
Local officials told federal emergencies minister Alexander Kurenkov that six people had died and six more were missing, Tass said. President Vladimir Putin ordered him to go to the scene shortly after the accident.
Footage from the site showed large parts of the nine-storey building on fire after the crash. Local authorities later said the blaze had been put out.
Krasnodar regional governor Veniamin Kondratyev denied local reports the plane had been loaded with ammunition.
"If that had been the case, then only half the building would be here," RIA cited him as telling reporters.
About 250 people were evacuated, including 40 children, authorities said.
RIA said the crash took place during a training flight from a military airfield. It quoted the defence ministry as saying the pilots reported an engine had caught fire on takeoff, and the plane's fuel had then ignited when it struck the building.
Russia's state Investigative Committee, which deals with serious crimes, said it had opened a criminal case and sent investigators to the scene.
The Kremlin said Putin had ordered all necessary help to be provided to victims. He also ordered the health minister to fly to the region.
The incident comes nearly eight months after Russia sent its troops into Ukraine.
