Hamburg SV and Standard Liege also won 3-1 while there were 1-1 draws between Rubin Kazan and VfL Wolfsburg, Valencia vs Werder Bremen and Benfica against Olympique Marseille plus a goalless affair in the Atletico Madrid vs Sporting tie.
The pressure on Liverpool manager Rafa Benitez continued after his side were undone by a fortuitous free-kick from teenage Belgium forward Hazard with just minutes left on a cold night.
The visitors, whose only chance of a trophy this season is in Europe's second-tier competition, started with an attacking lineup of Fernando Torres, Ryan Babel, Dirk Kuyt and Steven Gerrard while Lille were without their top striker Gervinho.
After shading the first half and having a close-range Torres header superbly saved by Mickael Landreau, Liverpool almost went behind after the break when Daniel Agger's attempted clearance narrowly missed his own goal.
Lille's winner came in equally bizarre fashion with six minutes remaining, when Hazard's long-range freekick evaded everyone including Liverpool goalkeeper Pepe Reina to hand the French side a slender advantage for the second leg.
"Lille are a good side, with good players and they are dangerous. We will play a difficult game at Anfield but have confidence we can beat them," the Spaniard told reporters.
The competition's other big name, twice European champions Juventus, eased past underdogs Fulham with first half goals from Nicola Legrottaglie, Jonathan Zebina and David Trezeguet.
The hosts, who surprisingly started with 23-year-old forward Antonio Candreva in favour of stalwart Alessandro Del Piero, went ahead early through Legrottaglie's powerful header and a superb Zebina strike.
Last season's beaten UEFA Cup finalists Werder Bremen drew in Valencia after captain Torsten Frings converted a first-half penalty to put them ahead. Juan Mata equalised just before the hour mark minutes after team mate Ever Banega was sent off.
