Northampton, who are guaranteed a play-off place, looked on course for victory until Collins sliced an attempted clearance over stranded goalkeeper Lee Nicholls four minutes from time.
Vale's vastly superior goal difference to Cheltenham (28 goals) means they will be playing in npower League One next season and that promotion left Adams emotional.
"I've been blubbering after the game, but now I've settled down I'm absolutely delighted for everyone connected with the football club," he said.
"We've been through the mill. Before the season started we were in administration and I was shouting down the phone at someone because we had no training kit for the players.
"What the players have achieved has been a minor miracle in the circumstances and I couldn't be more proud of them.
"We've had our doubters and cynics at various stages, but there has been a great spirit in the dressing room and the criticism brought us closer together.
"We never used excuses and the players have always stuck their chests out and gone out to do their best. I'm very proud of them and they should be very proud themselves.
It is the players who deserve most of the credit."