Ligue 1 leaders Lyon were sunk by a last-gasp Nicolas Fauvergue goal after twice cancelling out Lille's advantage.
Belgian 18-year-old striker Eden Hazard was the man of the match in Lille, scoring one and setting up the other two.
The home side went ahead on 22 minutes with Michel Bastos tapping in a cross from Hazard.
Cameroon midfielder Jean Makoun levelled two minutes later by heading home from a Miralem Pjanic corner. Hazard put Lille back in front on 44 minutes when he dribbled past four Lyon defenders to score but Lyon responded in stoppage time through Anthony Mounier's close range finish.
Neither side could find a breakthrough until, with extra time looming, Hazard ran past three Lyon players on the right flank before lofting a perfect cross for Fauvergue to score.
Paris St Germain, second in Ligue 1, suffered a shock exit as third division strugglers Rodez struck twice in extra time to go through to the quarter-finals.
Defender Jeremy Choplin gave Rodez the edge 24 minutes into extra time with a stunning 35-metre drive and striker Miguel Pacios put the result beyond doubt four minutes later with a clever volley from inside the box.
"It is huge, with the whole crowd behind us, we gave everything and we hoped until the end," Choplin, who also scored his side's opening goal, told Eurosport.
"We were a bit down after conceding the first goal but as soon as we equalised, we began to believe it was possible."
PSG started confidently and defender Sammy Traore grabbed the opener after nine minutes. The visitors looked set for a straightforward win but Choplin caught the PSG defence snoozing and raced into the area before beating Mickael Landreau from close range in the 66th minute.
Elsewhere, goals by Jimmy Briand, Romain Danze and Jires Kembo Ekoko gave Rennes a 3-0 win over Brittany rivals Lorient.
Monaco went through with a 2-0 win at Ligue 2 side Ajaccio and Ligue 2 Sedan beat fourth-division team Vitre 3-0.