The visitors had rarely threatened to find a way through after home defender Luke Prosser was sent off for a second bookable offence on the stroke of half-time. But Law raced into the danger area to curl a left-foot shot past goalkeeper Chris Martin from 15 yards to break valiant Vale’s hearts and earn Rotherham their first victory in five league games.
There were chances at both ends in a lively and hard fought first-half before Rotherham took a 31st minute lead. Right-back Dale Tonge ended a marauding run with a terrific cross which substitute Stephen Brogan headed high into the net from six yards.
Vale drew level in similar fashion five minutes later as Adam Yates whipped over a teasing cross from the right byline for the unmarked Prosser to head in.
But Prosser’s joy was short-lived as two bookings, for fouls on Nick Fenton and Adam Le Fondre in the 43rd and 45th minutes, led to an early bath.
Vale were content to defend in depth for much of the second-half and they did so to great affect with Rotherham rarely threatening a winner. In fact, it was Vale who came closest to going ahead before Law’s late strike with Marc Richards steering his shot a foot wide in the 55th minute.