It feels like a big win for Everton given the circumstances of their victory, though their fans will probably want it to be a little easier on the nerves in future.
As a fairly uneventful first half was about to draw to a close, some comical defending allowed the Baggies to go in front via Saido Berhaino, when chances had seemed to be few and far between.
But Everton managed to wrestle back the initiative almost immediately when Lukaku headed from a Gerard Deulofeu cross, and they started to grow into the game.
Lukaku was involved again as he turned provider for substitute Arouna Kone to equalise, and with five minutes remaining, Lukaku managed to finish from close range to give the Toffees all three points.
It means they now sit fifth in the Premier League after seven games, and have leapfrogged their near neighbours Liverpool which will give them some impetus going into Sunday's Merseyside derby at Goodison Park. It's a huge disappointment for West Brom, who are still without a home win during this campaign.
Here are the best images from last night's game: