After giving his side an early lead at Stadio San Paolo, Mertens won a second-half penalty which Jorginho converted to earn a commanding advantage ahead of next week's return tie.
Nice were without injured star Mario Balotelli and new signing Wesley Sneijder, and their relative inexperience showed as Mertens rounded the onrushing Yoan Cardinale for a 13th-minute opener.
The hosts spurned several opportunities but will still take a handy buffer to Nice, whose task grew tougher when young midfielder Vincent Koziello was shown a straight red for a studs-up tackle on Piotr Zielinski.
Celtic recorded the biggest ever margin of victory in the home leg of the play-off round in romping to a 5-0 thrashing of Astana at Celtic Park.
Tom Rogic forced an own goal on the half-hour mark and the home side never looked back, adding two more through Scott Sinclair either side of half-time before James Forrest and a second own-goal sealed the thrashing.
Sevilla are on track for a third straight appearance in the main draw after earning a gritty 2-1 away win against Istanbul Basaksehir courtesy of goals from Sergio Escudero and Wissam Ben Yedder either side of an Eljero Elia equaliser.
Croatian club Rijeka threatened an upset in Greece before Olympiacos rescued a come-from-behind 2-1 victory, their winner coming from a Jacques-Alaixys Romao header in the third minute of stoppage time after Vadis Odjidja cancelled out Brazilian forward Heber's surprise opener.
Hapoel Be'er-Sheva also mounted a 2-1 comeback win over Slovenian side Maribor to move closer to their first ever group stage appearance, Tony Nwakaeme's thunderous first-time volley cancelling out Marcos Tavares' goal before Shir Tzedek scored what proved to be the winner on the stroke of half-time.