However, the visitors pulled away in the second half, with goals from Tyrone Barnett, Grant McCann and Britt Assombalonga all securing the points, before Yoann Arquin netted a consolation from the penalty spot late on.
Both teams menaced throughout the opening exchanges, with Jamal Campbell-Ryce firing over for the hosts five minutes into proceedings, while Assombalonga came close for the visitors.
Magpies boss Chris Kiwomya watched on as his side monopolised possession, and he did not have long to wait to see the pressure pay dividends.
Captain Leacock rose highest to a Dumbuaya cross to thump home the opening goal of the game on 26 minutes.
The hosts were not ahead for long, though, as Posh captain Rowe provided an immediate riposte in similar fashion by heading home from a corner.
Assombalonga, who netted the winner against Swindon Town last week, thought that he had picked up where he left, but his goal was chalked off for offside.
Posh emerged from the break as the stronger side and took the lead on 54 minutes through Barnett, who slipped his effort beneath Bartosz Bialkowski.
Things got worse for the hosts four minutes later when McCann - a former Magpies loanee - scored a third from the penalty spot.
Posh midfielder Daniel Kearns spurned the opportunity to put the result beyond doubt 20 minutes from the end, but the 21-year-old blazed over from close range.
However, club-record signing Assombalonga was on hand to end the game as a contest moments later, capitalising on a defensive error before smashing past Bialkowski.
The hosts got a goal back in stoppage time through a Arquin penalty, but it would be little more than a consolation.