The Scotland international marked his 100th appearance for the east Lancashire club by scoring his 54th goal just hours after Rovers had issued a statement confirming they had rebuffed two bids from the Tigers.
Rhodes' opener kick-started a first-half blitz which saw the hosts score three times inside 11 minutes. Captain Grant Hanley headed in a second from Tom Cairney's corner and Rudy Gestede diverted Craig Conway's shot past Lee Camp moments later.
Blackburn were denied a first clean sheet of the campaign nine minutes from time, though, as Brett Pitman converted a penalty which Adam Smith appeared to have dived to win. Steve Cook then headed home in the 90th minute but Rovers were still able to seal a first home win of the season.
The Cherries were comfortable in possession in the early stages but lacked a cutting edge which Rhodes duly provided for the hosts at the opposite end.
The striker chased a long ball down the right and shrugged Cook off the ball before sliding his first chance of the afternoon between Camp's legs to give Blackburn a 13th-minute lead.
It was a poignant reminder of his importance to the team's aspirations of achieving promotion, while proving manager Gary Bowyer's assertion that Hull's interest would not disrupt him on the pitch.
Another two goals quickly followed as Rovers ran rampant. First, Cairney's corner was met by Hanley's head and the Rovers skipper's effort from near the penalty spot looped over Camp and into the net.
Cairney's next corner produced the third goal too; Conway collecting a half-cleared ball and seeing his shot turned in by Gestede's flick, with Camp stranded after committing to the initial effort.
A fourth goal nearly arrived prior to the break, Gestede bullying his way between two defenders to poke just wide while man-of-the-moment Rhodes struck the inside of a post following Markus Olsson's foray down the left.
The half-time whistle gave the visitors some welcome respite and they tested Paul Robinson with a flurry of chances at the start of the second period.
Yann Kermorgant's shaped shot was pushed over the crossbar by the former England stopper, who then twice denied Callum Wilson a fourth goal of the season at his near post.
With 13 minutes left Rhodes was taken off, clapping all four sides of the ground before embracing Bowyer, who had a lengthy chat in the striker's ear on the touchline.
He had been in his seat for just three minutes when team-mate Alex Baptiste gave away a penalty. Smith meandered through and hit the deck - although replays showed Baptiste's leg failed to make contact with his - and Pitman smashed home the resulting spot-kick.
Cook's downward header in the 90th minute put Ewood Park on edge but Rovers closed the game out, with the help of Baptiste's heroic block from Tommy Elphick deep into stoppage-time.