The Uruguay international took his tally to 15 in 19 Barclays Premier League matches. Firstly Suarez hooked a ball forward for Raheem Sterling to race onto and the 18-year-old coolly lobbed the goalkeeper.
James McClean should have equalised but he managed to miss the target with only Jose Reina to beat and it proved costly. From the hosts' next chance Steven Gerrard's pass released Suarez who was bounced off ball by Carlos Cuellar.
Referee Phil Dowd played a good advantage and the forward continued his run into the box before drilling a shot through the goalkeeper's legs. Reina then produced a brilliant stop to deny Matt Kilgallon.
But there was no way back seven minutes into the second half as Suarez struck for the second time. Inexplicably the second successive side to visit Anfield - Fulham were the other - decided the right tactic was to sit off Gerrard and let him have the ball.
He destroyed the Cottagers with his range of passing and the same fate befell Sunderland. The England captain collected the ball around the right-back position and, with no pressure on him, he had time to pick out Suarez's run 60 yards away.
His team-mate needed no second invitation and in collecting the ball he moved it to his right to unsteady Mignolet and fired home Liverpool's third.
Suarez was denied a hat-trick by the legs of the goalkeeper while substitute Joe Allen's shot was deflected over by Danny Rose.
Victory, which completed a haul of nine points out of a possible 12 over the festive period, lifted Liverpool into eighth place, seven points behind fourth-placed Chelsea.
A second successive defeat for Sunderland, after two wins, means Martin O'Neill's side are looking over the shoulders again, four points above the relegation zone.