Jennison Myrie-Williams fired home a 75th-minute winner after sub Ben Williamson - heading home in the 63rd minute from a Myrie-Williams cross - cancelled out Hourihane's brilliant 35th-minute solo goal.
Argyle had taken a second-minute lead through Argentinian winger Andres Gurrieri but Vale targetman Jordan Hugill made it 1-1 with an 18-yard strike after 29 minutes.
Grew said: "It is good to be in round four and we will look forward to the Brighton game (in round four).
"It was a little bit of deja-vu from the first game (which ended 2-2) where they caught us on the hop.
"The gaffer said at half-time that wasn't us and we were better than that. There was no shouting, just constructive advice on how we could get back into it. We showed what we can do in the second half.
"In truth they looked better on the ball than us in the first half and we were glad to come in 2-1 down.
"The lads showed great character in the second half and, in the end, we are all delighted to get through."