Kalinic twice gave Rovers the lead at The Hawthorns before being dismissed for a poor challenge on Paul Scharner shortly after Mame Biram Diouf added a third.
The Baggies had been level midway through an eventful Premier League encounter after Jerome Thomas cancelled out Kalinic's first goal, but later on their misery was compounded when Gabriel Tamas was also sent off.
Until his red card, the afternoon had looked like being one of Kalinic's best in a Rovers shirt since being signed by Sam Allardyce last year.
And the lively Blackburn performance eased fears the players do not want to play for Kean.
The visiting fans also voiced their support for Kean, although they were clearly not behind the club's controversial new Indian owners.
Blackburn also achieved their win without wantaway defender Christopher Samba who, after being stripped of the captaincy recently, failed a fitness test on an ankle injury.
The game got off to an explosive start with Blackburn claiming the lead after just three minutes.
Kalinic showed great pace as he raced on to a long ball down the left from the returning El-Hadji Diouf.
The Croatian then kept his composure as Scott Carson advanced and coolly slipped the ball past him low into the far corner.
Graham Dorrans had a similar chance to equalise at the opposite end after being put through by Peter Odemwingie but Paul Robinson got down to palm away.
Kalinic could have doubled Blackburn's lead after combining well with David Dunn on the edge of the area but his shot curled narrowly over.
West Brom responded to level with a superb goal from Thomas after 17 minutes.
James Morrison freed the dangerous Somen Tchoyi down the right and the Cameroon player delivered a perfect first-time ball into the area for Thomas to turn in at the far post.
The contest remained open and exciting and Blackburn attacked again down the left as El-Hadji Diouf fed Kalinic but Youssouf Mulumbu turned his cross behind for a corner.
Odemwingie then broke clear for the Baggies only to have a cross blocked at the expense of a corner.
The hosts finished the first half in the ascendancy as Michel Salgado was booked for a late challenge on Odemwingie.
The resulting Dorrans free-kick created some uncertainty in the Rovers box but the visitors hacked clear.
Blackburn were forced into a half-time change with substitute goalkeeper Mark Bunn coming on to replace Robinson.
Bunn was soon into the action as Odemwingie broke clear and tested him with a fierce shot which he turned behind for a corner.
Blackburn reclaimed the lead after 53 minutes as Kalinic grabbed his second.
Nelsen headed an El-Hadji Diouf corner back across goal and Kalinic jumped above the defence to nod home.
Kalinic almost pounced for a hat-trick moments later as Carson miscontrolled a backpass from Tamas but the goalkeeper recovered.
Another corner routine led to Blackburn's third just after the hour.
Pedersen took it short to El-Hadji Diouf and the Senegal international delivered a fine cross for namesake Mame Biram Diouf to head in from close range.
But just as Blackburn seemed to have taken a firm grip on the game, they found themselves a man down as Kalinic was dismissed.
The goalscorer tangled with Scharner just inside the hosts' half and referee Phil Dowd showed a straight red card after he crudely caught the Austrian below the knee with his studs.
That gave West Brom an incentive and they threatened to pull a goal back as Tchoyi again got wide to cross but Bunn produced a fine save to beat out substitute Simon Cox's first-time shot.
The Baggies came back again and Scharner produced a deft header that looked goalbound from a Dorrans free-kick but the ball rebounded off the crossbar.
It proved their last chance and their hopes evaporated when Tamas was dismissed for a professional foul on Mame Biram Diouf.