A witness told broadcaster TBC he saw the plane flying unsteadily as it approached the airport in poor visibility conditions, saying it took a turn for the airport but missed and went into the lake.
Video and pictures circulating on social media showed the plane almost fully submerged, with only its green and brown-coloured tail visible above the water line of Lake Victoria, Africa's largest lake.
Rescue boats were deployed, and emergency workers were continuing to pull trapped passengers from the plane, TBC reported.
Rescue workers were in touch with the pilots in the cockpit and were attempting to pull the plane from the lake, Albert Chalamila, the chief administrator of Tanzania's Kagera region, told reporters.
TBC footage showed scores of residents standing along the shoreline and others wading into the shallow waters, as rescuers carried on with their efforts.
Eyewitness Abdul Nuri - who had been at the airport awaiting the return flight - recalled seeing the plane plunge into the lake.
He told the BBC on Sunday: “We were really shocked. People panicked and some started crying and shouting.
“At the arrivals gate people panicked as well - most of them were waiting to welcome their relatives.”
Precision Air, Tanzania's largest privately owned airline, identified the aircraft as an ATR-48.
"We have managed to save quite a number of people," Kagera province police commander William Mwampaghale told journalists.
"When the aircraft was about 100 meters (328 feet) midair, it encountered problems and bad weather. It was raining and the plane plunged into the water," he said.
Tanzanian President Samia Suluhu Hassan called for calm as the rescue operation continued.
"I have received with sadness the news of the accident involving Precision Air's plane," she tweeted. "Let's be calm at this moment when rescuers are continuing with the rescue mission while praying to God to help us."
