Midfielder Yannick Ferreira Carrasco scored the winner four minutes from time after his team, who are second in the top flight standings, had struggled at the Rabine tadium.
Rodriguez levelled the tie again four minutes from halftime, cancelling out Yannick Aguemon's 35th-minute strike from inside the area.
Claudio Ranieri's side wrapped it up in the 86th minute when Ferreira Carrasco beat Antoine Petit with a powerful 18-metre shot.
They avoided the fate of Lorient, so far the only Ligue 1 team to be eliminated by an opponent from a lower division.
The Brittany side slumped to a 1-0 defeat at amateurs Yzeure, who play in the fourth division, on Saturday.
Paris St Germain's game at Stade Brest, which was scheduled on Saturday but was postponed because of bad weather, will be played on Tuesday (1915 GMT), the French champions said on Sunday.
Earlier on Sunday, Lille went 1-0 down at fourth division Amiens AC before Salomon Kalou's double and a goal by Ronny Rodelin handed them a 3-1 victory.