Hull, on the other hand, have won just one of their last 12 games and are without a point since drawing with Liverpool on 13 December.
Manager Phil Brown will be furious that Fellaini's goal was allowed to stand but could have no complaints at the result given Everton's superiority in a physical encounter.
Brown's men worked hard but were outclassed by an Everton side who started brightly and probed tirelessly for a breakthrough.
Despite the absence of a recognised striker, lone front man Tim Cahill was ably supported by Leon Osman, Steven Pienaar and Fellaini as the hosts dominated possession.
Hull's 4-5-1 formation indicated a determined rearguard action would be the order of the day and for 18 minutes they held firm impressively.
And in Marlon King, Bernard Mendy and former Everton forward Nicky Barmby - heckled throughout on his first return to Goodison Park having left the club for Liverpool in 2000 - they looked to catch the Toffees on the counter-attack.
Moyes pleased with 'professional' win
But their gameplan was breached when Baines linked well with Pienaar on the left and crossed for Fellaini, in a clear offside position when the ball was played, to nod past Boaz Myhill at the near post.
The home side could have doubled their lead in the 24th minute when Pienaar broke free on the right and squared dangerously across the face of goal, but none of his team-mates was on hand to convert.
Cahill then crossed for Fellaini to head narrowly wide and at the other end Michael Turner had Hull's only half-chance of the match, a tame shot dragged wide.
Everton were sitting comfortably on their lead but they were given the chance to double it when Turner fouled Cahill 25 yards from goal.
Arteta duly stepped up to rifle a blistering drive past Myhill, a wonderful strike and the Spaniard's third goal in two league games.
The second half was a scrappy affair and, with Everton not exactly over-exerting themselves in search of a third, underlined Hull's need for attacking reinforcements in the January transfer window.
Brown infuriated by 'offside' goal
Brown introduced Peter Halmosi, Craig Fagan and Daniel Cousin yet for all their graft the visitors did not create a meaningful chance all half.
In the 57th minute an outstanding tackle by Hull captain Ian Ashbee thwarted Osman when Everton had three men against one.
Victor Anichebe, on for Cahill, did little to press his case for a starting role and his one effort rolled harmlessly wide, while Fellaini received his 10th yellow card of the season, which will lead to a two-game ban, ruling him out of the Premier League and FA Cup meetings with Liverpool.
Osman shot high from a promising position late on but by that stage the points were in the bag.
Everton manager David Moyes:
"It is another win, another clean sheet and although Hull made it a scrap, we deserved the win.
"Referees we have had recently have been very good, but I don't think this one was. And we have had an offside goal, that was clear.
"I will be straight with things that affect both sides, and our first goal was offside. I have to be fair.
"The downside was the booking for Fellaini. I have seen it now on TV and the Hull lad ducked his head down to about four feet off the ground, Fellaini is looking at the ball and toe-pokes it away - it can't seriously be dangerous play, you are entitled to challenge for that."
Hull manager Phil Brown:
"Our whole season has been unhinged by a succession of officials and decisions we are not happy with.
"The first goal was a bad decision from an assistant referee who is looking right along the line of the defence.
"It killed us. We worked on not giving Fellaini space in the six-yard box, and we achieved that. But he was in an offside position in space we had left him in by squeezing out.
"Arteta's goal would have made it 1-0 at the break and we would still have been in the game. But I have players with blood all over their shirts, so they gave it everything in that second period."
Everton: Howard, Hibbert, Jagielka, Lescott, Baines, Osman, Arteta (Rodwell 90), Neville, Pienaar, Fellaini, Cahill (Anichebe 73).
Subs Not Used: Nash, Van der Meyde, Castillo, Jutkiewicz, Gosling.
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