Orient had originally gone 1-0 down in the first half, thanks to a goal from Posh midfielder Tommy Rowe in the 26th minute, but Mooney put the visitors on level terms just before the break.
The Os came out of the traps quickest in the second half as Dean Cox finished after a neat turn gave him space to fire home, before Mooney popped up again to secure his side's third of the afternoon and cement an important comeback win.
Wolverhampton Wanderers managed to keep the pressure on the league leaders as they notched up a 2-0 victory over Stevenage at Molineux, thanks to goals from Kevin Doyle and James Henry either side of half time.
Out-of-form Notts County continued their poor start to the season as they fell to the bottom of the table having lost 3-0 away at Coventry City, while Bristol City moved off the bottom into 23rd thanks to a 1-1 draw against Oldham Athletic.
Results in full: Bristol City 1-1 Oldham, Coventry 3-0 Notts County, Crawley 0-1, Brentford, Crewe Alexandra 0-0 Bradford, Gillingham 1-0 Carlisle, MK Dons 1-0 Walsall, Peterborough 1-3 Leyton Orient, Preston 1-1 Tranmere, Rotherham 2-2 Colchester, Shrewsbury 2-0 Sheff Utd, Swindon 5-2 Port Vale, Wolves 2-0 Stevenage