Assombalonga struck first in the 58th minute when he finished off Josh McQuoid's teasing low cross before slotting home from the penalty spot after Conor Washington was fouled.
The victory increased Posh's lead over MK Dons in the final play-off spot to six points - with Assombalonga and co also boasting a game in hand.
In the first half David Martin denied Assombalonga before diving at full stretch to tip Ajose's 25-yard shot wide.
At the other end, Ryan Hall, making his first start for the Dons, forced Bobby Olejnik into action with a 25-yard free kick.
Assombalonga missed a great opportunity to put the visitors ahead in the 57th minute when he headed over an open goal after Martin had saved well from Tommy Rowe.
But he made amends a minute later when he converted McQuoid's cross from five yards.
And two minutes later he scored from the penalty spot for his 27th goal of the season, sending Martin the wrong way after Anthony Kay pulled down Washington.