A dour game was lit up by the midfielder's unchecked run and shot into the far corner from the edge of the box after 57 minutes.
The first half had little to offer by way of entertainment as only Posh midfielder Grant McCann threatened goal when his free-kick from 20 yards hit the post but rebounded to safety.
Notts were the first to press after the break when Alan Gow's tame shot was saved at the near post by Joe Lewis.
Peterborough got their winner five minutes later when Lee raced from just inside the County half and when he was invited to shoot he fired past Stuart Nelson into the far corner.
Notts were a limp attacking threat as the closest they came to an equaliser was when Kevin McDonald saw his drive from 20 yards well saved by Lewis in the 82nd minute.