Rovers' hopes of reaching the Sky Bet Championship play-offs this season appear to be diminishing by the game after Grabban's 67th-minute strike consigned them to a third March loss on the spin.
This was a crucial fixture for Gary Bowyer's men after their East Lancashire neighbours won at Ewood Park three days ago, their first success in the derby for 35 years.
Bowyer had spoken of the need to avenge that defeat quickly yet instead their frustration now goes on, with the loss made all the more galling given former Clarets boss Eddie Howe is in charge of the Cherries.
Rovers had started promisingly and were on the front foot with the visitors pinned inside their own box in the early exchanges.
However, the first chance would fall to Grabban, sent clear when Blackburn skipper Grant Hanley misread Harry Arter's pass down the left flank. Grabban drew a smart stop from Paul Robinson and Hanley was back in time to block the rebound.
That prompted Howe's men to settle in to the contest and Matt Ritchie was the next to test Robinson, the former England stopper tipping his 25-yard try around a post.
Bowyer's side were struggling to build sustained periods of possession and their final pass was often found wanting. That was not the case when Rudy Gestede's improvised flick fell in to Luke Varney's path, yet the ex-Leeds forward stumbled and, in vain, claimed he ended on the turf via a Simon Francis shove.
The half ended with the Cherries on top during a succession of corners which began when Ritchie's shot hit Jason Lowe and swerved just wide of Robinson's left-hand post.
From the resulting corner, Ian Harte's deep delivery found an unmarked Steve Cook and he headed against the base of the post from six yards out.
Harte tried something different from his next set piece, teeing up Francis on the edge of the area with the defender's drive palmed over the top by the Robinson, who was probably among those happy to see the half-time whistle.
That was met by boos from some home supporters who had seen little to appease them after ceding the bragging rights to their neighbours for the first time since 1979 on Sunday.
Against such a tense backdrop, Howe's men were able to continue their fluid football after the break and only Corry Evans' brilliantly timed challenge on Ryan Fraser prevented the Cherries teenager bearing down on goal.
Moments later, though, Hanley's timing was less exquisite as he slid on Eunan O'Kane just inside the area and appeared to get nowhere near the ball, yet referee David Webb remained unmoved.
While Jordan Rhodes and Michael Keane headed off target, Blackburn were in need of inspiration so they turned to their favourite son, David Dunn, and loanee Liam Feeney from the bench.
However, just six minutes after that 61st-minute double change, the visitors had their opening goal. Grabban picked the ball up 25 yards out and Rovers defenders allowed him to set his sights on goal as he skewed a low drive inside Robinson's near post.
The stopper will surely feel he should have done better, but he was alert enough shortly afterwards as Cook waltzed through only to see his strike beaten wide.
Anxiety around Ewood only grew and, apart from a header which went over the top from Rhodes, the limp second-half showing only served to anger Blackburn fans further.
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- Eddie Howe
- Gary Bowyer