The South American pair made the difference in a match where the visitors had taken the lead through Juan Mata before squandering two more chances and putting under-fire manager Andre Villas-Boas under yet more pressure.
Lavezzi scored Napoli's first and third goals, the first of the competition for the Argentine and both laid on by Cavani who notched the other himself.
Napoli's win left England in danger of missing out on the quarter-finals after Arsenal, the Premier League's only other team in the last 16, were thumped 4-0 at AC Milan last week.
Asked if he expected to be in charge by the time his side host Napoli for the return leg on March 14, the Portuguese told Sky Sports: "Yeah, for sure."
Chelsea's day had got off to a bad start with news that captain John Terry could be sidelined for several weeks after aggravating a knee injury and it soon got worse.
"After the first goal, we missed opportunities to go 2-0 up and then we should have defended better but we made a couple of mistakes against Napoli with their attack you are going to get punished," said Villas-Boas.
"I think the third goal was perfectly avoidable... we can create problems with our attacking options and we created problems [for them] today.
"It is a game of efficiency and Napoli had it. I think the statistics are very similar for both teams. Both teams created a lot. The game was open."
They were fired up enough without him, although Lavezzi refused to get carried away, saying: "I was happy to score but the important thing was that the team won. We haven't qualified yet."
He saved a dangerous Cavani shot with his foot and then turned Christian Maggio's angled effort around the post as the Italians quickly showed their attacking potential with fast, pinpoint passing.
But the visitors went ahead with their first real chance after Napoli captain Paolo Cannavaro sliced Daniel Sturridge's cross and the ball fell at the feet of Mata, who beat Morgan De Sanctis with a first-time shot
Chelsea's rearguard got into another mess in first-half added-time as Gokhan Inlan delivered a teasing cross which bounced six metres from goal and Cavani stole in front of two defenders to fire Napoli in front with his shoulder.
