Paul Hayes scored the only goal of the game when he converted a penalty kick just before the break to seal a 1-0 League On win - welcome relief for Adkins after his side suffered defeat in their first three outings of the season.
"We've got the result we needed as a group", he told BBC Radio Humberside
"We stuck together as a group and the lads have worked extremely hard this week.
The United boss was slightly disappointed that his side did not take advantage of three excellent opportunities to increase their lead late on.
"We had opportunities to go and score another four goals but we did look a bit leggy. But we're observing that and addressing it.
"We've done it now, we've got a clean sheet and a well deserved victory and when the lads are back in on Monday we'll get ourselves prepared for two very tough away fixtures."
The Posh boss Darren Ferguson was unhappy with Kieran Charnock, who brought down Marcus Williams to give United the decisive penalty.
"I'm disappointed to go away with nothing" admitted the Scot.
"But I told the players that it's big decisions that win or lose matches and there was a big decision that was wrong a minute before half-time.
"It wasn't the referee that got the decision wrong though, it was Kieran Charnock"