Loan striker Sammon hit a double for the Blades, with Jose Baxter adding a late third as the Bramall Lane outfit produced the perfect riposte to their previous game on the road in League One - which ended in a humiliating 4-0 loss on the opening day of the campaign at Gillingham.
The win was United’s second on the spin in the league - and third on the trot in all competitions.
Adkins said: “I thought we started really brightly and Conor Sammon scored a great goal for us. The link-up play between Sharpy and Sammon looked dangerous.
“Conor scored two great goals and Sharpy could have scored a few as well.
“It is only the second game they have played together. It is a work in progress, but you could see with both of them with some of the link-up play straightaway looking for each other, which was great.”
On the overall display, Adkins, whose side welcome Blackpool at the weekend, added: “We wanted to attack and did that.
“The game was open and exciting; probably a bit too open. But credit to Peterborough because they made it made it that way as well.
“It was end to end and credit to them, they got themselves back into the game and could have had three or four goals.
“But we went and responded in the right manner and got ourselves back in front again.”