
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.
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.
In the other quarterfinals, Liverpool travel to Nottingham Forest or Huddersfield Town and Chelsea go to giant-killers Middlesbrough. Southampton host Manchester City in an all-Premier League clash.