The winning goal came in the 84th minute from Gary Sawyer after he capitalised on Paul Quinn's poor clearance.
Cardiff blew the chance to close the gap on leaders Newcastle and Nottingham Forest who both drew.
West Brom capitalised with a home win over City's next opponents, Peterborough.
Cardiff are still 4th but with a 14 point gap on leaders Newcastle and a 6 points gap to Albion who are 2nd.
Incredibly after 22 games played Cardiff have earned more points from their 11 away games than they have their 11 home games.