Yannick Noah's hosts went into the day needing just one victory from the final two singles rubbers to take an unassailable 3-1 lead against an under strength Serbia, who were missing Novak Djokovic, Janko Tipsarevic and Viktor Troicki.
Lajovic had stunned Lucas Pouille in the opening match of the last-four clash on Friday and he eased to the opening set, only for Tsonga to find his rhythm and eventually cruise to a 2-6, 6-2, 7-6 (7/5), 6-2 win.
Tsonga, the world number 18, was given some scary moments during the match, and had to save two vital break points with the third set locked at 4-4 before taking a tie-break.
But the former Australian Open runner-up took the fourth set by a double break as Lajovic's resistance was broken.
France are into their fourth Davis Cup final since the last time they won the trophy in 2001, and first since losing to Roger Federer's Switzerland in 2014.
They will face either Belgium or Australia in the final, with Lleyton Hewitt's visitors leading 2-1 in Brussels.