A first-half goal from Belgian winger Kevin Mirallas gave the Toffees the advantage before Diego Lugano's late header gave the Baggies a share of the points in Pepe Mel's first game in charge.
However, with 15 minutes remaining, veteran defender Lugano notched his debut goal for Albion when he met James Morrison's with a powerful diving header at the front post.
The result saw West Brom move up to 13th on the Premier League table while the Toffees dropped to sixth after Tottenham had beaten Swansea City on Sunday to move into fifth.
2013/14 Barclays Premier League season scores
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