That lifted Forest into 10th place in the table, while their opponents remain one off the bottom.
Davies was bullish about his side's progress after struggling on their return to the second tier last season.
He said: "When I came to the club in January I faced a massive job just to stay in this division. We managed to achieve that aim and this season is all about making progress.
"For weeks I have been telling anyone willing to listen about how well we have been playing but it wasn't possible to use our normal passing game in this match.
"It was all about scrapping and showing determination to get a result and I felt we shaded it over the 90 minutes."
Forest fell behind five minutes into the second half when Posh striker Mclean headed a Tom Williams corner past keeper Lee Camp. But they were not trailing for long.
Majewski conjured an almost immediate reply when he embarrassed substitute Posh keeper James McKeown with a curling free-kick from close to the left touchline.
And the transformation was completed soon after in the 56th minute when Anderson pounced from point-blank range to fire past McKeown after a left-wing Chris Cohen cross had been knocked down.
Defender Luke Chambers saw a looping header cleared off the line before substitute Garath McCleary struck the crossbar as Forest comfortably saw out the contest to leave their manager toasting the best run of his nine-month reign.
Lowly Posh's plight was hardly helped by the loss of two players to injury in the opening half an hour.
First-choice keeper Joe Lewis departed with concussion after a crunching challenge with Nathan Tyson and midfielder Paul Coutts soon followed him into the dressing room with a foot problem.
And Posh manager Darren Ferguson felt the loss of his keeper was a pivotal moment.
He said: "I certainly feel it affected the result. Joe couldn't see properly and has gone off to hospital - it would have to be serious to get him off.
"It is another game we did not deserve to lose but we are seeing all the time that small margins make a big difference at this level.
"We were guilty of over-celebrating our goal. It was as if we had won the World Cup and as a result we lost concentration and ended up with the ball in our own net less than 60 seconds later."