Everton, who will travel to Crystal Palace in the quarterfinals, made their Ukraine left back Vitali Mykolenko captain and fielded a much-changed team against the National League side. All four quarterfinals can be watched on ESPN+ in the U.S.
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They were frustrated until the 57th minute when Jonjoe Kenny slid in a low cross for Rondon to score at the near post, and the Venezuelan netted his second late on with a header.
Everton created little in the first half as their delivery from wide areas was poor. Mykolenko forced a good save from Taye Ashby-Hammond in the Boreham Wood goal and the keeper had to keep out a curling effort from the excellent Anthony Gordon.
The home side were brighter in the second half and Rondon had a simple headed chance from Kenny's delivery that he put over the crossbar, before the same two players combined for the opening goal.