The hosts took the lead early on as Mauro Formica was quickest to react to Rhodes's blocked shot to fire home in the fourth minute.
Rhodes got his first in the 19th minute, latching on to Martin Olsson's cross to head his side 2-0 up.
The Scotland international doubled his personal tally 20 minutes later, again beating the defence to a cross, this time provided by Ruben Rochina, to poke home.
Rhodes almost completed his hat-trick on the hour mark when he rounded the keeper, but the angle just beat him as he couldn't direct his shot on target.
Marcus Olsson came within inches of extending his side's lead even further with just over 10 minutes left, but his effort hit the bar.
They wouldn't have to wait long for their fourth, however, as Formica's corner was headed back across goal by Scott Dann for Rhodes to nod in his third of the match.
Morten Gamst Pedersen was denied the chance to make it five for Blackburn when his follow up to Martin Olsson's shot that hit the post was ruled out for offside.
There was still time for a consolation goal for the hosts, however, as Lee Tomlin lashed his shot home to make it 4-1.