James Beattie's first goal since his £6m move from Southampton helped Everton into the FA Cup fifth round.
James McFadden gave Everton the lead in the ninth minute when he flicked home Kevin Kilbane's cross at the near post.
And Beattie doubled the advantage after 27 minutes when he turned neatly on the edge of the area and saw his 12-yard shot deflected in off Danny Collins.
Tim Cahill rounded off Everton's win when he headed home McFadden's cross at the far post with 10 minutes left.
Beattie saw a header well saved by former Everton goalkeeper Thomas Myhre when he stole on to Kilbane's cross after eight minutes.
But it was a brief reprieve for the Black Cats as Everton went ahead seconds later.
Kilbane was again the provider with a dangerous cross that was turned past Myhre by McFadden at the near post.
Everton were dominating totally and it was no surprise when they doubled their advantage after 27 minutes.
Carsley played in Beattie, who turned neatly before seeing his shot from 12 yards deflected past Myhre by Collins.
Sunderland had barely been seen as an attacking force, but Carl Robinson provided a moment of anxiety for Everton six minutes before half-time when he volleyed narrowly wide from Julio Arca's cross.
And Arca produced the first serious save from Everton goalkeeper Richard Wright when he blocked his long-range shot.
Sunderland wasted their best opportunity to get back into the game after 69 minutes when Stephen Elliott cleverly worked space on the edge of the area, but sent his finish flashing past Wright but inches wide.
Everton sealed a comprehensive win after 80 minutes when Cahill - who scored the winner for Millwall against Sunderland in the FA Cup semi-final - headed home McFadden's cross at the far post.
Everton assistant manager Alan Irvine:
"It's Beattie's goal as far as we are concerned.
"We didn't even ask him because he is a striker. He will claim it and his performance deserved it.
"His shot was going in and it was on a good day for us. We are very happy with the performance and the victory."
Sunderland manager Mick McCarthy:
"We got a good hiding from a very good outfit. No complaints - we got a real good doing out there.
"But I've got to praise my players for keeping going. I'm full of admiration for them, but we were beaten by a side that showed why they are doing so well.
"I'll use the old cliche about concentrating on the league now, but I still desperately wanted to win."
Everton: Wright, Pistone, Yobo, Stubbs (Weir 65), Naysmith, McFadden, Carsley, Cahill, Kilbane (Bent 77), Beattie (Chadwick 77), Osman. Subs Not Used: Hibbert, Martyn.
Booked: McFadden, Naysmith.
Goals: McFadden 9, Beattie 27, Cahill 80.
Sunderland: Myhre, Wright (Brown 62), Neill Collins, Danny Collins, McCartney, Whitehead, Robinson, Thornton, Arca, Stewart, Elliott (Bridges 77). Subs Not Used: Caldwell, Lynch, Alnwick.
Booked: Wright, Elliott.
Ref: U Rennie (S Yorkshire).