The striker grabbed the only goal as the Robins were seen off at London Road in their Sky Bet League One curtain-raiser.
Assombalonga, 20, arrived for a fee in the region of £1.25 million from Watford earlier this week and needed just nine minutes to make an impact.
He held off defender Grant Hall and thumped a stunning finish past Swindon goalkeeper Wes Foderingham.
And it proved enough to earn Posh three points after both sides struck the woodwork in the second period.
Skipper Tommy Rowe blasted against a post as Peterborough searched for a second goal before Town debutant Ryan Mason hit the crossbar as they failed to grab a leveller.
"It doesn't get much better for a new striker than scoring the winning goal on his debut," said Ferguson.
"There is no doubt Britt is going to prove to be a great buy for the club.
"We saw that he is a real handful who will only get better.
"I'm really pleased to start the season with a win.
"We always knew this would be a tricky test and we're glad to get through it with three points.
"The opening day of the season is like the first day back at school - you just want to get it over with.
"It wasn't a great performance by any means, but it was solid and that is what is needed at this level.
"We were very good in the first half and could easily have been further ahead.
"We weren't as good after the break when Swindon came into the game more, but they never really looked like cutting us open.
"We know there is stacks of improvement to come."