“I think I could have had five in the end. I have to be more of a serial killer for the team to achieve their objective.”

Whilst the victory meant Ronald Koeman became the first Everton manager since Thomas McIntosh in 1919 to win his first two competitive away games for former Everton handler David Moyes it was sheer misery.

His Sunderland side have garnered just a point from the first four matches and have begun a season with four winless games for the sixth consecutive year.

Koeman, who arrived from Southampton in the summer, was furious with the first half performance but said he had not been worried by Lukaku’s lack of goals as he is a ‘great player’.

“I think we will be one of the teams that will fight for European football,” the Dutchman said.

“That is a realistic target. If we play like we did in the first half we will be 13th or 14th. The players are not stupid – they know what they can do.”

Moyes for his part said the new players he had brought in would take time to settle but that wasn’t a good enough excuse for their performance.

“I was really disappointed, especially with the last 30 minutes,” said Moyes.

Lukaku had the best chance of a frenetic first-half where both sides showed a willingness to attack but conjured up few opportunities.

The Belgian frontman, though, might have done better than heading it straight at young Sunderland goalkeeper Jordan Pickford, the England under-21 international still reacting brilliantly to tip the thunderbolt header over the bar.

Koeman made one change at the break taking off an under-performing Ross Barkley and sending on Spanish winger Gerard Deulofeu.

Sunderland were again on the back foot at the start of the second-half.

They lifted the pressure temporarily when central defender Lamine Kone – scorer of a brace when the ‘Black Cats’ trounced Everton 3-0 to ensure they remained in the Premier League in May – rose above Phil Jagielka but Maarten Stekelenburg was equal to the task of saving his 52nd minute header.

Sunderland escaped a minute later when the lively Deulofeu broke free but his whizzing shot was blocked by Jack Rodwell, only for the ball to work its way out to Seamus Coleman.

The Irish international’s cross shot sneaked across goal and although Lukaku got an outstretched boot to it it wasn’t enough to direct it into the net, the ball trickling wide of the post.

However, he made no mistake on the hour mark as unmarked he headed Idrissa Gueye’s cross past Pickford – Deulofeu the inspiration behind the goal having won the ball midway in his own half from Wahbi Khazri and bringing it deep into Sunderland territory.

The outstanding Gueye then went close after Lukaku scuffed his shot before Lukaku saw the crossbar deny him a second in the 63rd minute.

However, he was not to be denied five minutes later as Yannick Bolasie found space down the left, cut to the byline and crossed to an unmarked Lukaku who made no mistake with his header.

Two minutes on and Lukaku raced clear after being found by a superb slide-rule pass by Kevin Mirallas, and the Belgian striker slotted it nonchalantly past Pickford for his hat-trick.