Sunderland fought valiantly to the end but drew a blank again as Everton proved too hot to handle as they secured all three points at Goodison Park.
Bryan Oviedo and Darron Gibson – both January acquisitions – made quick returns to Goodison Park, while Jordan Pickford replaced Vito Mannone in goal.
As expected, in-form Everton started strongly with Pickford and his defence immediately under pressure.
The Sunderland keeper parried an Ademola Lookman effort after seven minutes, with the ball slipping behind him, but fortunately the keeper was able to recover in time to gather on the line.
Sunderland – with Adnan Januzaj and Fabio Borini supporting Jermain Defoe – responded as Oviedo threatened with a powerful drive which deflected wide off Seamus Coleman.
But it was Everton creating the better chances, as Pickford had to be alert to stop a low drive from Idrissa Gueye with his legs.
Having weathered the storm in the first 20 minutes, Sunderland settled into the game and the home crowd became restless at the extent of their side's progress.
However, the visitors suffered a huge blow just before the break as an swift Everton attack carved them open.
A ball in from the right ran across the box to the onrushing Gueye, who blasted a first-time shot into the top corner to leave Pickford with no chance.
Then Pickford was forced to turn away an awkward bouncer from Ross Barkley to keep the score to 1-0 at the break.
He got a bit of help from the post in the final moments – Tom Davies' rocket of a shot from outside the box rebounding back off the upright.
Sunderland were probably relieved to hear the half-time whistle as Everton enjoyed a strong spell.
The away side made a brighter start to the second half as both Defoe and Borini saw shots blocked, the latter after Januzaj seized on a defensive error.
And O'Shea flicked a Januzaj corner across a congested box, though Leighton Baines on the first post saw the ball comfortably wide.
At the other end, the Sunderland captain did brilliantly to flick a dangerous cross away from the lurking Lukaku.
But the visitors needed more in the final third, with Defoe isolated when they did break.
There was a definitely a sense of something in the game for Sunderland, though, with Everton making hard going of it despite their lead.
Sub Kevin Mirallas screwed an effort wide from a tight angle in a rare foray forward with 20 minutes to go with the game still in the balance.
Ross Barkley should have put the game to bed moments later when the ball dropped to him with the goal at his mercy, but somehow the midfielder planted his shot over the bar.
Pickford then came up with the goods courtesy of an excellent stop to deny Lukaku before Defoe came inches away from levelling.
A break led by Januzaj resulted in Defoe going one-on-one with a defender and letting fly with a shot which rattled the underside of the crossbar and bounced agonisingly down onto the line and away.
It was a missed chance which Sunderland would regret as, from the resulting corner, Everton broke away and Lukaku held off Billy Jones to score and secure all three points.
Everton: Robles, Schneiderlin, Baines, Williams, Barkley (McCarthy, 81), Gueye, Lukaku, Coleman, Funes Mori, Davies (Valencia, 69), Lookman (Mirallas, 60)
Subs not used: Stekelenburg, Jagiekla, Barry, Holgate
Sunderland: Pickford, Jones (Manquillo, 90), Kone, O’Shea. Oviedo, Borini (Khazri, 78), Larsson, Januzaj, Ndong, Gibson (Honeyman, 78), Defoe
Subs not used: Mika, Lescott, Love, GoochAttendance: 39,595