Jack Rodwell was sent off for a challenge on the Uruguay striker in the 23rd minute to leave the hosts playing with 10 men at Goodison Park.
Everton did well to hold its rival during a bruising contest. Toffees goalkeeper Tim Howard dived low to his left to save Dirk Kuyt’s penalty just before the break.
But the Reds broke through in the 71st when Jose Enrique picked out Andy Carroll inside the box for the opening goal.
Suarez ensured Liverpool’s second straight Premier League victory with eight minutes remaining after an Everton defensive mix-up gifted the ball to the Liverpool striker to score.