Zielinski claimed his second goal of the season with half an hour left at the Arechi Stadium as both sides finished with 10 men.
Napoli moved three points clear of second-placed AC Milan as Salernitana remained second from bottom with just two wins this season. Milan can draw level on points if they beat Jose Mourinho's Roma in the final game of the day. Napoli were without in-form Nigeria striker Victor Osimhen due to injury, with captain Lorenzo Insigne on the bench, and failed to create a clear first-half chance.
"Insigne has a muscular fatigue, he was at risk of getting hurt," Napoli coach Luciano Spalletti said. "I decided to keep him out, avoiding any risk," Spalletti added before Thursday's Europa League trip to Legia Warsaw.
Zielinski claimed the crucial goal on 61 minutes as he rifled home after a goalmouth scramble. Salernitana's Grigoris Kastanos was sent off with 20 minutes left for a rash challenge on Andre-Frank Zambo Anguissa. Napoli lost Kalidou Koulibaly eight minutes later as the Senegal centre-back was dismissed for denying Nigeria's Simy a goal-scoring opportunity.
Elsewhere on Sunday, Serbia attacker Dusan Vlahovic scored a hat-trick to take his goal tally for the season to 10 in 12 games as Fiorentina beat Spezia 3-0. Genoa and Venezia drew 0-0 while Polish midfielder Szymon Zurkowski opened his account for the campaign with a 92nd-minute winner to help Empoli edge Sassuolo 2-1.