Posh began confident of equalling a club record of 17 unbeaten league games but it was City who earned the plaudits with a sixth straight win.
Leicester took a fifth-minute lead when Matty Fryatt chased a seemingly lost cause down the right before crossing into the penalty area where Peterborough defender Craig Morgan sliced the ball into his own net from six yards out.
Peterborough came close to an equaliser on the half-hour mark when Aaron McLean struck a post from eight yards, and Craig Mackail-Smith crashed the rebound against the bar before McLean blazed the loose ball into the crowd.
Fryatt made it two for City after 38 minutes when he latched on to a Jack Hobbs long-ball and rounded Joe Lewis before guiding home an angled shot from 12 yards.
Leicester then added a third after 70 minutes when Matt Oakley played in a free-kick to the near-post with Andy King ghosting in beyond a static defence to score with a firm header. And Steve Howard sealed the victory after 90 minutes with a well-struck drive from 22 yards.