Stockport did pull one goal back through Liam Bridcutt in a game packed with chances, but Vale were full value for their victory.
Adams, who took Brighton to the Southern Section Final of the competition last season, said: "We are over the moon because we really needed the win. We have been playing well and not getting the results we have deserved.
"We could have won more comfortably, but Stockport also had their chances in what was a very entertaining game.
"I decided to play three forwards to make it difficult for their defenders to pass the ball out. Theories don't always bare fruit, but our plan worked. To see all three score was great. We've got competition for places up there and the players are responding to that, which is good for me as a manager."