N'Zogbia won the penalty that Christian Benteke converted to break the deadlock in the closing stages at Villa Park and the former Wigan winger then curled in a superb free-kick to give the hosts much-needed breathing space.
For Paul Lambert's Villa, it was the latest embarrassment in a disappointing season that has seen them thrashed 8-0 by Chelsea and slip to within one place of the Premier League relegation zone.
West Brom, meanwhile, have ended a run of three consecutive defeats, and the point secured by Odemwingie's goal was enough to take them level on points with both Arsenal and Liverpool in the Premier League.
Aston Villa and Wolverhampton Wanderers are unbeaten in the Premier League after the local rivals drew 0-0 on Saturday.
Aston Villa signed Wigan midfielder Charles N'Zogbia, while local rival West Bromwich Albion brought in goalkeeper Ben Foster on a season's loan as Midlands clubs geared up on Friday for the coming Premier League season.
Wayne Rooney scored twice as Manchester United remained five points clear at the top of the Premier League after a 3-1 win at home to Aston Villa