Albion slipped to second in the Championship after their first 0-0 of the season at Swansea.
Slaven Bilic's side draw a blank during a poor game with Jake Livermore having Albion's best chance late on to earn all three points.
Leeds' comfortable victory over Huddersfield saw the Baggies knocked off top spot with nine games of the season remaining.
Bilic made two changes to the side which lost to Wigan with Romaine Sawyers and Kyle Bartley restored to the starting XI.
The Baggies had just one shot on target in the first half with Callum Robinson's drive saved by Swans stopper Freddie Woodman.
Play-off chasing Swansea had opportunities of their own but never really challenged Sam Johnstone in the Albion goal.
The point keeps Albion six points clear of third-placed Fulham, after the White's draw at Bristol City.
Albion next face neighbours Birmingham City at The Hawthorns.