The hosts were looking to regain some momentum but instead stretched their winless run to five games.
The Cherries rushed out of the blocks despite a 30-minute delay to kick-off, with both goalkeeper Artur Boruc and skipper Tommy Elphick held up in traffic.
Bournemouth winger Marc Pugh repeatedly combined with full-back Charlie Daniels but Eddie Howe’s side struggled to carve out many clear-cut openings.
Daniels had been restored to the side by Howe after dropping to the bench in their 2-1 defeat to Nottingham Forest.
Pugh volleyed over after seven minutes while Rovers defender Matt Kilgallon blocked Daniels’ early strike.
Bournemouth were largely frustrated by the visitors and were limited to shooting from distance – including Harry Arter’s low drive after 38 minutes.
Blackburn top-scorer Rudy Gestede went closest to breaking the deadlock on 42 minutes for Rovers when he flashed his header narrowly wide from a Craig Conway corner.
The Cherries did not give up in search of a late winner and striker Yann Kermorgant was unfortunate to see his effort cannon back off the crossbar.
The French striker latched onto a cross from Simon Francis but his effort rebounded off the woodwork.
Kermorgant was at it again when his header was blocked from a Ryan Fraser corner.
Pugh was rampant throughout but his tame effort after 54 minutes was comfortably dealt with by Steele.
Rovers never really looked like scoring until boss Gary Bowyer threw on Jordan Rhodes to liven things up.
Rhodes ran down the left before picking out Gestede but he looped his header wide of the goal.
The Bournemouth defence had to be on their toes and both Elphick and Steve Cook ensured Gestede’s chances were kept to a minimum.
Howe sent on Matt Ritchie to conjure up some magic and he instantly connected with Wilson but Rovers defender Alex Baptiste intercepted after allowing the 18-goal striker to jink into the box.
Substitute Tom Cairney gave Bournemouth a late fright when he ran past their midfield before sending a dipping shot just wide of Boruc’s post two minutes from time.