Danny Murphy added a second six minutes later when Cool fumbled a long-distance shot.
Gerard Houllier's side now face Celtic in the next round.
Liverpool: Dudek, Carragher, Henchoz, Hyypia, Traore, Diouf, Murphy, Gerrard, Riise, Owen, Heskey. Subs: Arphexad, Baros, Smicer, Hamann, Diao, Biscan, Cheyrou.
Auxerre: Cool, Radet, Boumsong, Mexes, Jaures, Faye, Tainio, Mathis, Lachuer, Fadiga, Cisse. Subs: Hamel, Grichting, Gonzales, Mwaruwari, Perrier Doumbe, Sirieix, Akale.
Referee: A Lopez Nieto (Spain)
