Rhodes headed in goals in both halves, as well as a coolly-taken finish, while Mauro Formica's first goal since April set the ball rolling.
Lee Tomlin's late goal was scant consolation for Peterborough who have lost for the fourth time this month and slump to the bottom of the Championship.
Blackburn struck early thanks to a devastating counter attack. Rhodes wriggled into space in the box and his shot found Formica via a deflection, the Argentinian calmly slotting in from six yards.
Rovers grew in confidence and almost got another after 13 minutes when some silky passing ended with Ruben Rochina curling a left-foot shot from 20 yards that Robert Olejnik had to push behind for a corner.
Darren Ferguson's side did not heed the warning signs and were made to pay again after 20 minutes as Martin Olsson sent in a left-wing cross that Rhodes headed home from six yards despite Olejnik's best efforts, though there was a hint of an own goal after the ball hit the hand of Nathaniel Knight-Percival before going in.
Peterborough had no answer to the former Huddersfield striker who struck again before half-time. Rochina's delightful through-ball sprung the offside trap and found Rhodes in the box who confidently stroked his shot into the near corner.
The home side needed a fast start to get anything out of the game and Saido Berahino tested Paul Robinson in the opening minutes of the second half with a fiercely swerving effort that the former England man pushed over the crossbar.
But Rovers still posed a threat and Rhodes went close to his hat-trick after being played in by Formica. He rounded Olejnik but placed his shot wide.
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, and they almost had a fifth after Rochina's 25-yard shot was deflected onto a post.
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 Rochina and then managed to divert a Nuno Gomes rebound onto the woodwork.