Blackburn went behind after just five minutes when Mark Davies found space to fire home and Nigel Reo-Coker made it 2-0 for the visitors on the half-hour after more sloppy defending.
Amid chants of "Kean out" from the disgruntled home fans Blackburn responded with a Yakubu Aiyegbeni goal after 67 minutes but they could not find the equaliser and now prop up the table on 10 points from 17 matches, two behind Bolton.
In Tuesday's other game, lowly Wolverhampton Wanderers salvaged a point against promoted Norwich City in a 2-2 draw thanks to a late header from Ronald Zubar at Molineux.
The result left Norwich ninth with 21 points from 17 games, while Wolves crept up one place to 16th with 15 points, two clear of Wigan Athletic in the relegation zone.
FANS PROTEST
Blackburn fans made their opinions heard after the match by staging a protest, though Kean denied the club's Indian owners have called a meeting to discuss his future.
"That report is not correct, there's nothing in place at the moment," the Scot told Sky Sports.
"I've not got a date yet where I'm going to meet the owners, but when that comes around we'll speak about the transfer window.
"We've got a lot of defenders who are injured at the moment, so we've got to grind through it because as soon as we get them back we'll go on to win games."
Kean's counterpart Owen Coyle, himself under pressure following a poor first-half of the season, expressed his sympathy for the Blackburn boss after a topsy-turvy game.
"Blackburn Rovers are a fantastic club. The supporters pay their money but have they ever given Steve a chance from the outset? I don't think so," Coyle said.
The pressure is only likely to mount for Kean given Blackburn's next two matches are against Liverpool on December 26 and Manchester United five days later.
At the other end of the table, leaders Manchester City welcome Stoke City and United play Fulham on Wednesday, while third-placed Tottenham Hotspur take on fourth-placed Chelsea on Thursday.
