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The Manchester United forward clinically side-footed the ball home at the back post in the 54th minute at Leicester City's King Power Stadium after an over-hit corner was swung back in by Kyle Walker.
It was Rashford's fifth goal for England in 27 matches.
The World Cup semi-finalists, who made nine changes to the side beaten 2-1 by Spain in the Nations League at Wembley on Saturday, had provided few thrills in a first half controlled by the more flowing Swiss.