Shrewsbury debutant Gozie Ugwu and Vale striker Ben Williamson both threatened in the lively opening stages, but Vale wrestled the initiative before taking a 23rd-minute lead.
Shrewsbury felt they should have had a penalty when Adam Reach went to ground from Robertson's 30t- minute challenge, but the visitors were in deep trouble when Vale scored again just seconds later.
Jennison Myrie-Williams burst down the right before cutting inside to curl a brilliant left-footed shot high inside Weale's right-hand post from 25 yards.
A poor first-half display by Shrewsbury was summed up when Liam McAlinden wasted a golden chance to reduce the arrears.
McAlinden did make amends in the 59th minute when his left foot slammed a loose ball inside Chris Neal's right-hand post from 10 yards.
And Shrewsbury performed much better after that with Vale having to dig deep before Hugill clipped a left-footed shot past the advancing Weale in the fourth minute of time added on.