It was Becchio who gave the home side the lead on seven minutes, finding space inside the box to pick up a Luke Varney pass and fire into the bottom corner.
Leeds almost doubled their advantage on 27 minutes when McCormack whipped in an inviting cross for the Argentine, but his header flew over the bar.
Becchio had to wait until the second half before bagging his second goal. The striker capitalised on poor defending from Posh to nod home McCormack's cross on 50 minutes.
Leeds looked in control for a while after the second goal went in, but were guilty of complacency on 73 minutes, allowing Michael Bostwick to pounce in the box and pull a goal back.
However, Bostwick's strike proved no more than consolation for Darren Ferguson's side as Leeds tightened up and held firm until the final whistle.