Cunha doubled the lead for United in the 56th minute when he fired a pass from Casemiro into the roof of the net from close range.
The result moved Liverpool up to fifth on the EPL table.
Liverpool returned to winning ways in the Champions League on Wednesday night, beating Marseille 3-0 in France in their penultimate match of the first round.
City controlled 67 per cent of the ball and registered 14 shots, yet could not breach a resolute Sunderland defence on a freezing New Year’s Day in the North East.
Sunday’s intense Premier League clash saw Aston Villa host Manchester United in a must-win game for both sides.
Newcastle next face Manchester United away, while Chelsea host Aston Villa next Saturday.
The final whistle confirmed Arsenal’s dominance as they remain top of the Premier League standings.
After a wasteful first half with both teams missing several chances to break the deadlock, Barnes opened the scoring for the hosts in the 63rd minute.
Just as Tottenham thought they had nailed the victory, de Ligt headed home a corner kick from Bruno Fernandes to secure a draw for Man United.
Hazard joined Chelsea from French Ligue 1 side Lille during the 2012 summer transfer window.
