Chris Dagnall had put the Tykes ahead but a straight red card for Jean-Yves M'voto moments later and a Tom Cairney equaliser from the resulting free-kick turned the tide and once Rhodes had struck his first on the half hour, the result was never in doubt.
Tomasz Cywka's emphatic strike was of little consolation to Barnsley, who are without a win this season, and they have tasted victory just once in their last 12 in all competitions.
Both teams made one change each from their last league games, with Josh King replacing Chris Taylor for Rovers and Kelvin Etuhu in for David Perkins for Barnsley.
Rovers started brightest, but Barnsley took the lead after 13 minutes, when Dagnall was allowed time and space on the edge of the area to turn and shoot and his precision strike nestled into the bottom corner.
The pendulum swung right back in the favour of Rovers three minutes later when M'voto cynically brought down Rhodes, leaving the referee with no option but to show a red card.
And Cairney delivered instant retribution with a wonderful free-kick that found the top corner of Luke Steele's goal.
Sustained pressure followed, with King denied by Steele who had initially flapped at a cross, and then Tom Kennedy wore one in the face to thwart Jason Lowe's goalbound effort.
The pressure told after 30 minutes and it was the impressive Kane who swung in a deep cross for Rhodes to expertly send a looping header past Steele, and he put the result beyond doubt nine minutes later with a cool lob from a Scott Dann long ball that caught Steele in no man's land.
Rovers were rampant and got their fourth after 43 minutes when David Dunn sent Kane clear who provided his second assist of the game. This time King was the beneficiary and he tapped in from six yards.
The Tykes came out looking to restore pride and new recruit Marcus Pedersenn almost did that but was thwarted by Grant Hanley and Rovers netted a fifth through the marauding Kane after 58 minutes. It owed much to a howler from Steele who somehow managed to palm his shot into his own goal.
Barnsley did get a second after 72 minutes when Pedersen's audacious flick found Cywka who lashed in a sumptuous effort from 25 yards but it was of little consolation for the beleaguered Flitcroft.
- Sports & Recreation
- Tom Cairney
- Todd Kane
- Chris Dagnall