Foden has 24 goals for City in all competitions this season and has often been utilised in more central roles by coach Pep Guardiola
After Palace's Andros Townsend rattled the bar with a thundering effort from distance the visitors were awarded a penalty when Mamadou Sakho tripped Mohammed Salah.
Despite Palace's protests, Milner kept his cool to fire past Wayne Hennessey and send Juergen Klopp's side into the halftime break with the advantage.
Palace's task was made harder 15 minutes from time when defender Aaron Wan-Bissaka was sent off for bringing down Salah in a goalscoring position.
Mane made it 2-0 in stoppage time with great strength and skill as he broke from the halfway line, after collecting a Salah through ball, and rounded Hennessey before slotting home for Liverpool's second win in their two games this season.