The Tonga international, 30, joins up with international team-mates Nili Latu, Sonatane Takulua and Taione Vea at Kingston Park.
Since moving to the Premiership from Bayonne in 2015, Fonua has scored 16 tries in 42 domestic and European games for London Welsh and the Tigers.
"Opeti is a big physical specimen," said director of rugby Dean Richards.
"He gives us a different dimension. He is already experienced in Premiership rugby and he is looking good."
Fonua, who played for his country at the 2015 World Cup, added: "Things are in place for the Falcons to have a big year.
"The fact that many of Tonga's best players are already in the squad here was a big part in my decision."
Newcastle Falcons have signed back-rower Opeti Fonua from Leicester Tigers on undisclosed terms.