The son of Sir Alex, a former Manchester United player himself, saw his side run the Rovers ragged in their second-round clash at London Road.
Record signing Britt Assombalonga helped himself to a perfect late birthday present with the first hat-trick of his career a day after turning 21.
And striker partner Shaun Jeffers toasted a first Posh start by grabbing his first two goals for the club when striking in stoppage time at the end of each half.
The result was never in doubt after Posh, who also won 5-0 at Tranmere in a league clash in August, turned their domination into goals late in the first half.
Ferguson said: "Hopefully we get a good draw now. Manchester United away is the one I want and it would be ironic to get it this year.
"Financially it would be brilliant for the club to get a trip to one of the top Premier League clubs and it would give the lads the opportunity to test themselves against the best."
Reflecting on the game, Ferguson added: "I'm a positive person and have always believed in these players. Perhaps it just needed a result like this to get the lads' confidence up again.
"Our success over the years has been down to playing with a lot of energy and we had an abundance of it here.
"It was an excellent all-round display but the two lads up front did very well.
"Britt finds space brilliantly and his finishing is ruthless while Shaun has been chomping at the bit and deserved his chance."