Flight PW494, operated by Precision Air, hit the water during storms and heavy rain, the state Tanzania Broadcasting Corporation (TBC) reported.
Rescuers in boats rushed to the wreckage, which was almost fully submerged, to pull out trapped passengers, local authorities said.
"All Tanzanians join you in mourning these 19 people... who have lost their lives," Prime Minister Kassim Majaliwa told reporters in the lakeside city of Bukoba, close to the scene of the crash.
Investigators were still looking into what happened, he added.
The plane left the commercial capital Dar es Salaam and "crash-landed" at 8.53am on Sunday as it was approaching Bukoba airport, Precision Air - Tanzania's largest privately owned airline - said in a statement.
The plane was carrying 39 passengers, including an infant, as well as four crew members, the airline added.
It initially said 26 of the 43 people on board were rescued but later said 24 survivors were reported by emergency services at the scene.
A witness told 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 on social media showed the plane almost fully submerged, with only its green and brown-coloured tail visible above the waterline of Lake Victoria, Africa's largest lake.
Footage from the broadcaster and onlookers showed scores of residents standing along the shoreline and others wading into the shallow waters to try to help pull the aircraft closer to the shore with ropes.
Rescue workers were initially in touch with the pilots in the cockpit, Albert Chalamila, chief administrator of Tanzania's Kagera region, told reporters.
The prime minister later said the pilots may have died.
Precision Air identified the aircraft as an ATR42-500.
The Franco-Italian manufacturer ATR did not immediately respond to a request for comment.
First introduced almost 40 years ago, the ATR42 is the smaller of two series of short-haul turboprops made by ATR, a joint-venture of Airbus and Leonardo.
The last fatal accident was in 2017, according to safety database aviation-safety.net.
Tanzanian President Samia Suluhu Hassan called for calm.
"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."
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