Alvaro Morata found the net twice in three minutes to complete the comeback after Joaquin Correa put Lazio in front and Adrien Rabiot scored a screamer.
The Aquile had got off to the perfect start in Turin, Correa pouncing on a Juan Cuadrado error, but Rabiot surprised Pepe Reina at the near post with a ferocious finish to equalise.
After Sergej Milinkovic-Savic saw his header come off the crossbar, the game changed, as Federico Chiesa sent Morata clear to score, then the Serbian brought down Aaron Ramsey for a penalty.