David Wheater put Boro ahead on 44 minutes, but in the second period Everton overturned their one-goal deficit when Marwan Fellaini and Louis Saha scored on 50 and 56 minutes respectively.
Out-of-favor midfielder Mohamed Shawky was not included in Middlesbrough's squad.
The Egyptian international has been kept on the sidelines for a while, which prompted him to reveal his desire to move away from the Riverside by the end of the season.
However, coach Gareth Southgate didn't change his attitude towards Shawky, and assured the former Ahli player is expendable.
Arsenal Through
Earlier on Sunday, Arsenal qualified for the semi-finals after beating Burnley 3-0.
Carlos Vela broke the deadlock on 25 minutes, before Eduardo da Silva added the second nine minutes from the hour mark.
Seven minutes from time, Emmanuel Eboue put the icing on the cake with a powerful shot.
Previously, powerhouses Chelsea and Manchester United booked berths in the FA Cup semi-finals.
The draw for the last four takes place on Sunday night.