Thankfully, the Toffees did not have to wait long to find the back of the net after the restart. Three minutes into the second half, James Garner slid George in through the left channel and, when the winger's shot squeezed through the hands of Hornicek, Barry was left with the simple task of tucking home from a yard out.
With the wind in their sails, Everton went close to a second as Branthwaite's header was turned over the bar by Hornicek, before the Newcastle keeper did the same with Dewsbury-Hall's whipped shot from 25 yards out.
And from the resulting corner, Garner picked out O'Brien at the back post to nod down a goalbound header which Hornicek did well to turn behind.
Having done little to worry Jordan Pickford up until the hour mark, the hosts when so close to levelling when substitute Yoane Wissa nodded into Anthony Elanga at the back post who squeezed a finish under the England No.1 but, thankfully, the ball rolled along the line before being booted clear by Tarkowski.
As the inevitable Newcastle pressure grew trying to find a leveller, substitute Jacob Ramsey hit a volley from the edge of the area with 11 minutes remaining that crept inches past the left-hand post.
Then, as eight minutes of injury-time approached, Wissa looked to apply a finish from close range only to be denied by the legs of Pickford.
And that was to prove crucial as Everton saw out the closing stages, Ramsey being given his marching orders with seconds remaining after picking up two quick yellow cards for late tackles of substitute Hackney and Armstrong.