Chelsea rode that tricky start and began to create chances.
Bertrand Traore, making only his third league start for Chelsea because of an injury to Diego Costa, tested Mignolet.
Then Pedro – playing in a face mask after breaking his nose in an accident at home – wriggled through, and was denied by a terrific Kolo Toure challenge, before Mignolet pushed aside a long-range Hazard shot.
LIVERPOOL (4-2-3-1): Mignolet; Clyne, Toure, Lovren, Moreno; Can, Milner; Lallana, Coutinho, Firmino; Sturridge. Subs: Bogdan, Skrtel, Smith, Lucas, Allen, Ojo, Benteke.
CHELSEA (4-2-3-1): Begovic; Azpilicueta, Mikel, Cahill, Rahman; Fabregas, Matic; Pedro, Willian, Hazard; Traore. Subs: Amelia, Tomori, Kenedy, Loftus-Cheek, Palmer, Pato, Abraham. Referee: Michael Oliver.