Napoli dominated throughout a rainy fixture at the San Paolo but struggled to pierce a stubborn Lazio defence until Higuain scored with a deft touch in the 82nd minute to send Rafael Benitez's side through.
Defender Christian Maggio whipped a dangerous looking ball into the penalty area from the right and when Lazio defender Giuseppe Biava tried to clear, the ball fell to Jose Callejon on the edge of the area.
The Coppa Italia semi-finals are held over two legs. Udinese, who stunned AC Milan in the quarter-finals last week, host Fiorentina on 4 February before travelling to Florence for the second leg a week later.
Napoli will travel to Roma's Olympic Stadium on 5 February before hosting the Giallorossi a week later.