The Australian had a fine game for the Championship underdogs on Sunday but ruined his performance by allowing an Alexis Sanchez shot to squirm under his body for the winner.
Sanchez's effort slipped through the 30-year-old's hands and although Federici lunged back to collect the ball, it was already a metre over the line.
"Just like to say thank you for all your messages and how sorry I am for my mistake," Federici said on his Twitter account on Monday.
"We gave our all and deserved to at least take it to pens (penalties). It's part of being a goalkeeper and I'm sure it will make me stronger. Thank you again for your amazing support."
Arsenal's victory means they will meet Aston Villa, who beat Liverpool in the other semi-final, in an all-Premier League showpiece match at Wembley on May 30.
(Writing by Tony Jimenez, editing by Ken Ferris)
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