The visitors had defended resolutely until Mead’s introduction as a second-half substitute.
The Arsenal forward scored the first just moments after coming on to the field, with a brilliant finish on the turn past goalkeeper Jacqueline Burns.
Another substitute, Bethany England, scored the Lionesses’ second, with a touch on the line to ensure the ball went into the back of the net.
Mead then scored at the back post before tapping the ball into an empty net in the 78th minute to bring up her hat-trick.
England go top of Group D with maximum points from three games on goal difference ahead of Austria.
The Lionesses were playing a competitive fixture for the first time at the new Wembley.
But it was viewed by a crowd of just 23,225, significantly lower than the 30,000 the English Football Association had been expecting for the fixture.
England will host the women’s European Championship next year with the draw taking place in Manchester on Thursday.