The hosts fell behind in the 52nd minute when they failed to properly deal with a corner and James Morrison volleyed the ball past Paul Robinson.
Scott Dann scrambled in an equaliser with 18 minutes to go, but Blackburn were left empty-handed by Odemwingie's delightful 89th-minute effort.
This was a game Kean had had high hopes of taking all three points from, but it was a disappointing performance from his team, who remain 19th in the Premier League.
The Scot did not have to endure organised protests at a home match for the first time in a few months, but Rovers fans still made their unhappiness known, as chants of 'Kean Out' rang out around the ground after Odemwingie's goal.
The Baggies, meanwhile, are up to 13th and were deserved winners on the day.
Kean made two changes for the contest, midfielders Steven Nzonzi and forward Ruben Rochina - returning from suspension and a hamstring complaint respectively - coming into the first XI, with defenders Gael Givet (heart problems) and Michel Salgado (rib injury) missing out after both came off in last weekend's 2-1 defeat at Sunderland.
Albion boss Roy Hodgson made one adjustment from his side's 2-1 home loss to Wigan, with Paul Scharner replacing the benched Jerome Thomas in midfield.
The first chance of the game came to Odemwingie, who was released on the left by Morrison, but having cut into the box, he blazed the ball wide.
Odemwingie's accuracy was also lacking on the quarter-hour mark as he glanced Chris Brunt's cross wide when he really should have done better.
Two minutes later Ayegbeni Yakubu almost found himself in on goal at the other end, but could not stretch enough to make meaningful contact with the ball, and Odemwingie then tried his luck again, firing in an effort which Robinson turned around the post.
The hosts won a free-kick on the edge of the box which was curled goalwards by Morten Gamst Pedersen and tipped over the bar by Ben Foster.
Brunt then delivered one for West Brom, Shane Long getting his head to it but sending the ball too high.
Blackburn survived a scare shortly after the restart when Scharner sent a shot bobbling inches wide of Robinson's post.
Their relief was short-lived though as an Albion corner came off Nzonzi to Morrison, who cracked a volley in from just outside the area to make it 1-0.
Kean shuffled his pack by bringing on Jason Roberts and Grant Hanley for Simon Vukcevic and David Dunn, a move that drew cries of 'you don't know what you're doing' from some Rovers fans.
Dann then got away with an error of judgement, inadvertently lobbing Robinson when trying to play the ball back to his goalkeeper and grateful to see the ball bounce wide.
The former Birmingham defender made a far more positive contribution in the 72nd minute as he drew Rovers level, Christopher Samba heading on Robinson's long kick and Dann was on hand to bundle home.
There was another nervy moment for Blackburn as the ball ricocheted around their box, but Robinson pounced on it and Morrison then drove over the bar.
David Goodwillie, who had come on for Rochina, hit a shot straight at Foster, and Nzonzi and Yakubu were also denied.
Odemwingie then delivered the late hammer blow, dancing around several men before curling a beauty into the corner of the net.
This was the first of a vital double-header of home games for Blackburn and Kean will need something better from his team when they host Bolton on Tuesday if his reign at the club, which began 12 months ago, is to go on much longer.