Having completed a 10-game ban for biting Chelsea’s Branislav Ivanovic last season, Suarez scored in each half at the Stadium of Light as Liverpool ended a run of three games without victory.
Daniel Sturridge had given Liverpool the lead with his fifth goal of the season, with Emanuele Giaccherini replying for Sunderland early in the second half after Suarez had made it 2-0.
It was Sunderland’s first league game since the dismissal of manager Paolo Di Canio, but although they are now four points from safety, interim coach Kevin Ball will have been heartened by their second-half reaction.
Both sides threatened from early set-pieces, Keiren Westwood denying Liverpool captain Steven Gerrard and Sebastian Larsson rattling the visitors’ bar, before the away side went ahead in contentious circumstances in the 28th minute.
A left-wing corner found Sturridge unmarked at the back post and although he shaped to head the ball, it actually struck his elbow before bouncing over the line.