Premier League: Everton 2 West Brom 1
The hosts looked to be cruising to victory after Leighton Baines plundered two first-half goals, but substitute Shane Long ensured a nervy finish with a headed goal for West Brom on 65 minutes.
Both sides started tentatively, with missed headers from Phil Jagielka and Romelu Lukaku in a cagey opening 10 minutes.
Matters took some time to improve, with 18 minutes having passed before Marouane Fellaini shook Ben Foster's post with a half-volley inside the penalty area.
It would take a further 10 minutes for another example of genuine quality, but this time it would prove more decisive.
Baines collected the ball from Leon Osman 25 yards out after one of his trademark surges up the left and found the top corner beautifully from the edge of the box.
West Brom were dealt a further blow when George Thorne was forced to leave the field with injury.
Matters got worse when Thorne's replacement Claudio Yacob tripped Anichebe in the box to hand a penalty to the hosts two minutes after coming on.
Two minutes into first-half stoppage time, Baines subsequently stepped up to send Ben Foster the wrong way and give his side a 2-0 half-time lead, which they had earned without breaking a sweat.
The teams came out with notably more enthusiasm in the second period but this simply resulted in a scrappy midfield battle rather than added quality, meaning Everton looked certain of closing out an easy victory with an hour gone.
However, the introduction of Long for the ineffective James Morrison changed things significantly, as the Irishman capitalised on John Heitinga's error in the box to nod home with virtually his first touch.
That goal prompted a late flurry of attacking surges from both sides, with Osman's terrible miss in the box the most notable. The ball had bounced fortunately for the one-time England international, but he did not take the invitation and rifled over from just six yards.
That was the best opportunity either side had before the end, as Moyes' men saw the game out well to earn another important win.