Vale responded in the 29th minute when targetman Jordan Hugill scored with a superb low drive into the corner from 18 yards following a one-two with Louis Dodds on the edge of the box.
Argyle's response was swift as Hourihane drove forward on a defence-splitting run from the halfway line before smashing a 25-yard shot on the run into the bottom corner after 35 minutes.
Vale levelled when substitute Ben Williamson headed home a cross from Myrie-Williams in the 63rd minute before the latter cracked in the winner.
Micky Adams' side will now host Brighton in the fourth round.
- Sports & Recreation
- Chris Neal
- Conor Hourihane
- Chris Lines
- Jordan Hugill
- Reuben Reid
- Lewis Alessandra