Home skipper Antony Kay forced the ball home in the 93rd minute after an Ian Moore strike had bounced back off the post.
Nicky Forster hit the side netting inside two minutes for Albion, while at the other end Chris Shuker put a free-kick narrowly over the bar and then flicked another effort inches wide.
Tranmere made much of the running and visiting keeper John Sullivan did well to push out a powerful strike by Kay before Chris Greenacre headed just wide five minutes before the break.
As the tempo quickened in the second period Steve Jennings, Kay and John Johnson went close for Tranmere.
At the other end Forster brought an excellent stop from Danny Coyne before hitting the outside of a post after 61 minutes.