An early red card for Charlie Barnett gave Vale a great chance to leapfrog Gillingham and move back to the top of the table.
But Accrington, who have now won just one of their last 14 games, were the better team until home debutants Lee Hughes and Darren Purse scored early in the second-half.
Veteran Hughes completed the scoring with a third and Adams said: “Our first-half performance was very disappointing.
“Too many of our players were having too many touches on the ball and running in straight lines, which made it relatively easy for Accrington.
“But we had a good talk at half-time to sort a few things out. Our wingers came inside more so we could get our full-backs forward in the second-half and we started to create chances.
“Our midfield two of Ryan Burge and Doug Loft were also much better with their work and thankfully we got the job done and it’s another three points.
“I also had a word with Lee Hughes at half-time because I felt he was snatching at a few things and wasn’t as clinical as I’ve seen him – certainly when he’s played against teams I’ve been with in the past.
“But one thing you know about Lee is if he keeps getting chances he will soon start to score goals. He took those two chances very well.”
St Johnstone manager Tommy Wright has accused Neil McCann of deflection tactics as he denied reporting the Dundee boss to the Scottish Football Association.