It was the away side who started the better and they moved ahead after just five minutes when Marek Hamsik found the top corner with a curling effort.
Napoli might have extended their lead after 22 minutes when Edison Cavani came close with a strike, but he just missed the target.
Leece were not without their chances, however, and David Di Michele took one with an effort after 26 minutes in the match.
Napoli extended their lead after 51 minutes, however, when Cavani converted from close range to leave Lecce with a mountain to climb.
Daniele Corvia came close with an effort for Lecce after 69 minutes, but his effort was blocked.
Chances remained few and far between in the latter stages, with Napoli holding on for a vital win in their push for Champions League football.