Roma defender Anthony Rudiger was repeatedly subjected to racism from Lazio fans during the first half.
The chanting died down after the stadium announcer warned that the tie would be interrupted if it continued.
Sergej Milinkovic-Savic and Ciro Immobile scored as Lazio beat their rivals for the first time since 2013.
Lazio went ahead when Felipe Anderson burst down the right and pulled the ball back for Milinkovic-Savic to fire into the roof of the net.
Italy forward Immobile then tucked home their second from close range.
The second leg takes place on 5 April.
Juventus beat Napoli 3-1 in the other semi-final first leg.
Lazio established a two-goal advantage in the first leg of their Coppa Italia semi-final with Roma in a derby marred by racist chanting.