Despite this wretched run of form, Stoke could have scored only a 10th away league strike in the opening minutes when Walters tapped onto the woodwork from five yards after Tim Howard did well to push out a Robert Huth header.
An uninspired match then came to life when Mirallas beat Steven Nzonzi to Howard's punch from a Stoke corner, danced past his retreating opponents at pace, and effortlessly tucked the ball in for his sixth goal in an injury-hampered
first season on Merseyside.
The visitors were the first side to threaten in a dull opening after the interval, but defender Huth glanced Glenn Whelan's inswinging free-kick wide as he failed to properly connect with the ball at the back-post.
Jolted into life, David Moyes's men picked up the pace once more and went close with two threatening quick-fire free-kicks on the edge of the Stoke penalty area from specialist Leighton Baines and striker Jelavic.
With returning defender Phil Jagielka at the fore, the Toffees then professionally saw out the final stages to claim a victory that left them dreaming of Europe and their opponents dreading a relegation fight.
Stoke manager Tony Pulis:
"You cannot blame the players. Their attitude and application has been absolutely first class.
"It is our fifth year in the Premier League, and it is our best in terms of performances away from home, we just have not got the results. It is a strange game in lots of respects but we have to keep on believing.
"It is difficult times, but it is times you get paid for. I thought we played some smashing football today, and we do not get credit at times for the football we play."
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