Cavani double dumps Inter out of Italian Cup
The Uruguayan striker converted from the spot early in the second half after being fouled by Thiago Motta, linked by his agent with an imminent move to Paris Saint-Germain.
Cavani scored his second on the counter-attack in injury-time.
Hosts Napoli dominated the first half with Christian Maggio and Walter Gargano going close while an otherwise lethargic Inter had a good shout for a penalty turned down shortly after Cavani's spot-kick when Maggio challenged Diego Milito.
Napoli will play Siena over two legs in February and March after the Tuscan side overcame Chievo 1-0 away thanks to Mattia Destro's 54th-minute strike.
Siena were also reduced to 10 men on the hour when Paolo Grossi was dismissed in a game that was briefly threatened earlier in the day by a minor earthquake in northern Italy.
Serie A leaders Juventus beat AS Roma 3-0 in Tuesday's quarter-final and will next meet the winners of Thursday's encounter between AC Milan and Lazio.