Everton eased to a 3-0 victory at Swansea on Saturday, with
scoring to maintain the
club's fine start to the Premier League season.
Anichebe fired home from close range in the 21st minute, and Kevin Mirallas finished off a superb flowing move just before half time.
Swansea's hopes of a comeback were thwarted early in the second half when substitute Nathan Dyer was sent off, and Fellaini headed in Everton's third in the 82nd.
Everton has 10 points from a possible 15 so far this season, while Swansea is three points further back.
Such was Everton's dominance during the first half, it was a surprise it took the visitors so long to open the scoring.
Swansea's sloppiness on the ball and shoddy defending meant goalkeeper Michel Vorm was forced to make three saves inside the first three minutes.
Everton's opener came after Ashley Williams misjudged a high ball, allowing Fellaini to control and tee up Anichebe to score from 10 yards.
Swansea finally managed to find a spark, with record signing Pablo Hernandez bursting down the right before his chip was headed off the line by Phil Jagielka . Everton captain Phil Neville t hen blocked Danny Graham 's strike after Wayne Routledge had seized the loose ball.
Ki Sung went closer still for Swansea, clipping an effort just wide from 25 yards with goalkeeper Tim Howard beaten.
But Everton then re-asserted its authority with a wonderful goal.
Neville beat Michu to a long ball near his own right corner, starting a move that resulted in Fellaini and Steven Pienaar combining superbly to tee up Mirallas. Goalkeeper Michel Vorm did well to palm the initial strike onto the crossbar, but the Belgium forward was back on his feet quickly to convert the rebound for his first Premier League goal.
Angel Rangel had two chances to pull one back for Swansea before the break. A low shot was saved by Howard, before Rangel inexplicably volleyed Routledge's cross away from goal.
Laudrup withdrew Hernandez and put on Dyer at the break and the home side increased the pressure.
Ki was denied by Howard's save and the American goalkeeper also blocked Michu's effort.
But Dyer saw red just 12 minutes into the second half. The winger had already been booked for dissent when a clumsy tackle on Leighton Baines earned him a second yellow.
Dyer's dismissal sapped the life from what had been an entertaining contest, and Fellaini wrapped the scoring up as he headed home from Baines' cross with the aid of a deflection off Williams.