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The visitors came out of the blocks at a blistering pace and it paid off straight away, as Lukaku capitalised on a Jonas Olsson mistake to rifle home from 18 yards with only two minutes on the clock.
Albion looked rather toothless in the opening half despite sharing some encouraging possession, and Craig Gardner could only send his long-distance effort over the crossbar. The midfielder later sent a free kick from a promising position straight into the Everton wall.
The Toffees could have extended their lead against the run of play as the half drew to a close, when Kevin Mirallas found space before sending his shot just a little too high over the bar from 20 yards out.
Lukaku was unable to continue his efficiency early in the second period, when the striker's shot from 22 yards was kept out by Ben Foster, before Steven Naismith blasted over from the centre of the box.
West Brom's keeper suffered a howler in the 66th minute, when Mirallas decided to have a tame pop at goal, and Foster somehow allowed the ball to squeeze underneath him, handing Roberto Martinez's men a two-goal lead.
Aiden McGeady sought to put the icing on the cake for Everton and broke forward to find Leon Osman with just over 10 minutes to go, but the midfielder saw his attempt kept out of the top corner by a diving Foster.
Albion are still looking for their first victory of the season and under new boss Alan Irvine.