The striker scored just 22 seconds after the break to condemn Sunderland to a fifth straight defeat and ensure Liverpool will finish the year seventh in the Premier League.
Benteke should have made it 2-0 late on when he was denied by Vito Mannone while Roberto Firmino hit the post for the visitors in the first half.
Sunderland worked hard but struggled to create any clear chances, with Fabio Borini going closest when he shot narrowly wide in the 33rd minute.
The defeat means Sunderland end the year second from bottom in the table, seven points from safety.
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