Rhodes scored twice to take his tally for the campaign to eight in nine matches as Blackburn, in the Sky Bet Championship match, surged into a 3-0 half-time lead and then held off Bournemouth's second-half rally.
Bournemouth had substitute Richard Hughes sent off for two bookable offences in four second-half minutes, but rallied to score a late consolation through Ryan Fraser.
Blackburn went ahead in the 10th minute when Rhodes, with his left hand still heavily strapped after the injury he sustained on Scotland duty, dived full length to divert Leon Best's left-wing centre past goalkeeper Ryan Allsop.
Best, making only his second start of the season, added another in the 24th minute, after Corry Evans had found him free in the box to score from 12 yards.
Hughes came on for Shaun MacDonald, who went off with a knee injury in the 32nd minute, before Bournemouth went further behind four minutes before the break.
Josh King, who was a constant threat on the left, was fouled inside the area by Tommy Elphick to earn a penalty, and Rhodes stepped up to send Allsop the wrong way with his spot-kick.
Hughes received his first yellow in the 46th minute for a foul and was dismissed by referee Rob Lewis four minutes later for a late challenge on Evans.
Down to 10 men, Bournemouth began a belated revival with Marc Pugh hitting the bar in the 57th minute and Fraser racing clear in the 74th minute to fire home Tokelo Rantie's pass for his first goal for the club.
Bowyer said: "Jordan has great ability in front of goal and is a natural finisher.
"His (first) goal was stunning and brave because he was prepared to put his head in where it hurts, bearing in mind he has his arm in plaster.
"We were outstanding in the first half, excellent. I'm delighted for the fans because it is a long way to come and we have given them an away win to go home with.
"We've spoken a lot about scoring the first goal, which is something we've not done until today, and we carried out the game plan perfectly."
Bournemouth manager Eddie Howe questioned the mentality of his team after the opening goal, and judged it a match they need to consign to history smartly.
Howe said: "We were beaten by the better team and on occasions like this you have to hold your hand up and admit it.
"They have some clinical players who showed in the final third how dangerous they can be. We were not at our best in the first half and it would be easy to feel sorry for ourselves, but we have an important game at Leeds on Tuesday and have to put it behind us.
"We can do a lot better than that. We lost our belief when we went behind, which is strange for us, but in the second half that was more like the Bournemouth team I have been used to watching.
"Blackburn were excellent, but what pleases me is that we had a right go in the second half and scored a great goal.
"What was frustrating was that by then we were down to 10 men and left ourselves with far too much to do.
"Richard Hughes should not have received his yellow card for the first offence, I thought the referee had a poor game.
"But I think Richard will regret his challenge for his second card and left us in a vulnerable position."