Murphy converted a cross from Craig Alcock following a slip from Hereford centre-half Martin Riley.
But Yeovil were lucky to have 11 players still on the pitch after Carl Dickinson raised his hands towards Jacob Jagger Cane. Dickinson escaped with a yellow card.
Hereford almost found themselves 2-0 down shortly after Murphy scored, but Daniel Odo headed over.
Prior to those incidents, it was very quiet first half.