Mauro Formica's first goal since April set the ball rolling and Rovers were running wild as Rhodes struck his first two efforts.
First he headed home from six yards despite Robert Olejnik's best efforts then he confidently stroked into the near corner.
Berg's side continued to push for their fourth, first Markus Olsson struck the bar, before Olejnik got down well to push Martin Olsson's drive over the bar.
Their pressure was rewarded from the resulting corner, which was headed into the six-yard box by Scott Dann for Rhodes to nod in unchallenged for his ninth career hat-trick.
Peterborough grabbed a consolation shortly before the end when Kane Ferdinand squared the ball to Lee Tomlin who swept home and unstoppable shot into the top right corner, but Berg's side almost grabbed another. Olejnik saved brilliantly from Ruben Rochina and then managed to divert a Nuno Gomes rebound onto the woodwork.