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 skipper Rowe provided an immediate riposte in similar fashion by heading home from a corner.
Assombalonga, who netted Peterborough's winning goal against Swindon Town last week, thought that he had picked up where he left off, but his goal was chalked off by the referee for offside.
Both sides had opportunities to add a second during the remainder of the half, but both goalkeepers were equal to the efforts.