Posh put on a brave fight but Bricketwood set a new club record with four wins in a row.
Alex Tovey put Posh ahead with a strike from close range after a corner. Bricketwood equalised through Ben Kelly before Cameron Cockerill put the ball into the top corner from 20-yards to give Posh the lead again.
Tommy Hobson pulled on back for the visitors before Posh conceded a penalty and Kelly go this second of the match.
Both sides could have had more. Liam Avey hit the post twice and new goalkeeper Michael Easton was kept very busy by Bricketwood.