The home side were dominating and Kean was tested twice in quick succession when Scott Arfield's low shot was pushed away to Danny Ings, whose volley was beaten out by the keeper.
Ten minutes before the break Blackburn had their best chance of the game when Tom Cairney saw his shot saved by Tom Heaton.
At the end of the first half Ings had a chance for the home side but could only strike his shot from the edge of the area just wide.
Blackburn nearly opened the scoring midway through the second half when Grant Hanley headed onto the bar after a succession of corners for the visitors.
Junior Stanislas put the home side into the lead in the 75th minute when he played a one-two on the edge of the area with Ings, before firing in a low shot from 20 yards into the bottom corner that gave Kean no chance.
Blackburn were level 10 minutes later through Jordan Rhodes after Michael Duff's clearance cannoned back of the striker and looped over keeper Heaton.
In the closing stages substitute Lee Williamson was sent off for Blackburn less than two minutes after arriving on the pitch for his cynical challenge on Ings, but the visitors held on to claim a valuable point against their Lancashire rivals.