One Sheffield side had already fallen to Vale in the first round of the cup and last night Vale ran out 2-0 winners against the other Sheffield side, Wednesday.
Stunning goals from Kris Taylor and Robert Taylor secured victory for the League Two side.
Adams, who supported United as a boy, said: "I didn't like knocking United out but beating Wednesday was a different matter.
"Even if we play Manchester United in the next round, this is the tops for me.
"I know people will say we beat 10 men and of course that was a factor but I thought the attitude of my players was spot-on from the start. It would have been a travesty if we had not gone through.
"The game plan was to give Wednesday as little time on the ball as possible and I thought our tactics were spot-on. Every player did their bit and it's not a time to single out any individuals."