Arsenal's Alexis Sanchez snatched his second goal at the end of the first extra period to subdue the plucky second-tier strugglers, but his tame goalbound shot barely trickled over the line after being spilled by the devastated Federici.
It was an unfortunate way to decide a pulsating cup tie in which Reading's spirit and sheer will to win allowed Garath McCleary to equalise for Steve Clarke's side early in the second half after Sanchez had given Arsenal a 39th-minute lead.
Arsenal, who were playing in their record 28th FA Cup semi-final, will now face either Liverpool or Aston Villa, who play their last-four clash on Sunday, in the final on May 30.