The Southern League Division One South and West side, appearing at this stage for the first time, trailed at the break to goals from Chris Birchall and Myrie-Williams.
The latter then scored again from the penalty spot before Chris Lines headed home in stoppage-time to secure Vale a second-round clash at home to Salisbury City.
Shortwood stated their intentions to attack when Duncan Culley broke down the left and forced a corner inside the first 10 minutes but, four minutes later, they were behind.
Birchall cut in from the left wing and fed fellow winger Myrie-Williams who took one touch and finished across goalkeeper Tom King.
Birchall doubled the lead 15 minutes later, tapping in from close range after King denied Tom Pope.
Tim Haddock stung Vale goalkeeper Chris Neal's hands early in the second half but the visitors should have added a third when Birchall opted to dummy rather than shoot from 10 yards out.
Pope then headed against the crossbar yet Vale did add another after 89 minutes when James Coates was penalised for handball and Myrie-Williams crashed the ball into the top corner before Lines headed in a fourth.
- Sports & Recreation
- Chris Birchall