Central striker Marc-Antoine Fortune was also a key player with his hold-up play while Morrison had another effective game in midfield. But Albion controlled all departments and snapped away at the challenges in midfield, not allowing Martin O'Neill's side to settle.
O'Neill, who is 60 on Thursday, will hope this is a one-off performance after transforming Sunderland's fortunes and only Craig Gardner posed any sort of goal threat.
Odemwingie needed only three minutes to break the deadlock. Youssouf Mulumbu supplied a teasing cross from the left of the Sunderland box and Odemwingie got in front of Kieran Richardson to send his glancing header past Simon Mignolet.
It was the Nigerian international's nine goal of the campaign.
Albion dominated possession and midfielder Andrews tested Mignolet with a dipping drive.