The tie was as good as over after Crewe took charge with two goals by Rodney Jack in the space of three first-half minutes, with Vale never looking like getting back into it.
Crewe manager Dario Gradi said: "Rodney and David were the best two players on the field. They were the difference between the sides.
"When you've got two players who sparkle like they did, and everyone else does their job, you will win games."
Vale manager Brian Horton said: "I hate to say it but Crewe were the better side and deserved to go through. They wanted it more and passed the ball very well.
"We have to be a lot more positive at home and our second half performance was very disappointing. The goals killed us and we should have had the character to try to get back at them."