The striker came close to a second minutes leader when his free kick forced a save from Bournemouth goalkeeper Ryan Allsop.
Blackburn were dominating and doubled their lead in the 24th minute. Corry Evans was played through and unselfishly squared for Best to tap in.
The away side made it 3-0 four minutes from the break through Rhodes, who scored from the spot following Tommy Elphick's foul on Best.
Bournemouth's chances of getting back into the game suffered a blow just minutes into the second half when Richard Hughes was dismissed after a second bookable offence.