Steve Clarke's side put the absence of Nicolas Anelka behind them to earn a merited away draw after the former France forward threatened to retire earlier this week following the death of his agent.
Everton - who remain unbeaten after drawing at Norwich City last week - had plenty of pressure and possession, but they failed to seriously test the visiting goalkeeper Ben Foster, who was forced to depart injured late in the match to be replaced by Luke Daniels.
Marouane Fellaini came closest to a winner for Roberto Martinez's men late in the game when he controlled a Phil Jagielka cross, but his shot came back off a post from eight yards out with Daniels beaten.
But Tim Howard also had to look lively in the home goal to make saves from Markus Rosenberg and Shane Long in the second period.
'7 - PIENAAR TO THE FORE: Early chances for the Everton midfielder, but Ben Foster is alert to make timely stops from a header and a shot.
30' - EVERTON CONTINUE ON THE OFFENSIVE: Coleman picks out Mirallas. A powerful drive by Mirallas, but Foster is alert to nudge that one over the bar.
54' - FOSTER SHOWS HIS CLASS: Excellent stop by Foster from the Barkley drive before he gets back on his feet to save from Fellaini.
60' - EVERTON KEEP PRESSING: Another chance for Everton, but Mirallas can't beat Foster with the drive after giving Ridgewell the slip. Plenty of venom on the shot, but not taken.
72' - WEST BROM IMPRESS ON THE BREAK: Howard dives to his left to save from Long after he was played in by James Morrison.
74' - FOSTER FORCED OFF: Luke Daniels takes over from Foster, who is obviously injured. Big blow to West Brom, who have defended stoutly today.
80' - LAST CHANCE FOR HOME SIDE: Feillaini hits the post from a Jagielka delivery. Controlled the ball with his chest, but the shot comes back off the post.
88' - EVERTON ALMOST GET LUCKY: Coleman fires in a cross. Comes back off the West Brom bar. Almost a fortunate winner for Everton.
TALKING POINT: How are Everton shaping up under Roberto Martinez? Bit too early too say, but there seemed to be a reluctance to throw bodies forward late in the game with West Brom fairly comfortable in defence. One suspects these sides won't be too far apart in the table when the season ends.
Everton kept a clean sheet for the sixth Premier League match in succession at Goodison Park – their best run of clean sheets at home between November 1987 and March 1988, when they went eight home matches without conceding (top flight club record).
West Brom failed to score in their opening two matches for the first time in 13 years – since failing to score in their opening two Div 1 matches in August 2000 (1-0 at Nottm Forest and 2-0 home v Bolton). They last did so in the top-flight 33 years ago when they lost 1-0 home v Arsenal & drew 0-0 at Stoke, August 1980.
Everton open a season with back-to-back draws for the first time since the old Division One in 1970/71. They finished that season in 14th place.
MAN OF THE MATCH
Jonas Olsson (West Bromwich Albion) - Fine performance at the back to keep Everton's forwards in check as the visiting side held out with some ease in the end.
Everton: Howard 6, Baines 7, Jagielka 6, Distin 6, Coleman 6, Osman 6, Fellaini 7, Pienaar 7, Barkley 6, Mirallas 5, Jelavic 5; Subs: Kone 6, Naismith 6.
West Bromwich Albion: Foster 7; Jones 7, McAuley 7, Olsson 8, Ridgewell 7; Morrison 7, Mulumbu 7, Yacob 7, Dorrans 7; Long 7, Rosenberg 7; Subs: Brunt 6, Daniels 7, Vydra 7.
THE MATCH IN A TWEET