Adams returned to the dug-out after a three-game absence following hip replacement surgery and a very tight contest looked set to end in stalemate when big striker Barnett struck five minutes from time to keep Peterborough hot on the heels of leaders Leyton Orient.
Substitute Danny Kearns played the vital pass and Barnett found a yard of space to crack an unstoppable 10-yard shot inside Vale goalkeeper Chris Neal's right-hand post.
Adams said: "The effort and application from our players was magnificent and I feel gutted for them.
"We decided to match up Peterborough's midfield diamond because they've got some very talented players.
"We got in their faces and won tackles and I was very pleased.
"If I was to be critical we could have been better going forward.
"We need to keep the ball better. We tried to play too many killer passes instead of being a bit more patient, but that is something we can work on.
"It was a great effort from us from one to 11. Peterborough are a very good side, but standing on the touchline it looked like we were as good as them.
"To end the game with nothing is very disappointing, particularly for the players.
"We were so close to a great point when a bit of quality from Barnett and a bit of slack defending from our point of view has made the difference."