Injury-jinxed Louis Saha reminded everyone at Goodison Park what a fine finisher he can be when he is able to get on a pitch.
The Frenchman has been out since November with a hamstring problem, but, after coming on as a second-half substitute, he produced a stunning finish on the turn from nearly 30 yards to settle the points for Everton.
Saha has now scored four in 13 appearances and this was full of genuine quality.
Earlier Tim Cahill had scored his eighth of the season to put Everton ahead at the break and, despite a spirited fightback, Albion are left rooted at the bottom.
With Mikel Arteta, Victor Anichebe and Jack Rodwell all added to their injury list, Everton were happy to welcome back Tony Hibbert and Tim Cahill from suspension.
Moyes also restricted Joleon Lescott to the bench, because if he had picked up a booking, he would have been suspended for the FA Cup quarter-final against Middlesbrough on March 8.
But Everton suffered yet another injury problem soon after. Hibbert looked to have damaged a thigh early on, and after 26 minutes he was replaced by Leon Osman.
Albion were at this stage causing problems but all their industry went to waste when they conceded a shocking goal. Jay Simpson's horribly misplaced pass resulted in a free which saw three Everton men get away from their markers, with Carson hopelessly exposed as Cahill powered in a free-header.
Albion pressed on regardless and did cause problems but after 70 minutes Saha looked to settle the issue when he took a flick from Pienaar, turned and drilled an outstanding 25-yard drive inside the far post.
Osman then tested Carson with an angled drive, with Saha's fine goal having knocked the wind out of Albion's sails.
Despite another injury crisis, Moyes worked his magic once more. But the question is how long he can do so.