The 20 year-old striker, snapped up from Watford for a club-record fee earlier this week, needed only nine minutes to open his account.
Assombalonga held off defender Grant Hall to reach a long ball from Michael Bostwick and thundered an unstoppable shot past Robins keeper Wes Foderingham from just inside the box.
That strike ensured a losing return to London Road for Robins caretaker boss Mark Cooper.
Assombalonga could have helped himself to a second goal moments later but scooped his finish over the bar following a knock-down from strike-partner Tyrone Barnett.
Skipper Tommy Rowe was closest to doubling Posh’s lead in the second half when firing a low drive against a post.
Swindon, beaten play-off semi-finalists last season, were also denied by the frame of the goal when loanee Ryan Mason – one six debutants in the Town starting line-up – struck the bar late on with a 20-yard effort.