Athletic had led 1-0 from the first leg of the semi-final but fellow La Liga side Granada levelled the tie on the night through Carlos Fernandez in the 48th minute with a fine header from a cross by Venezuelan Darwin Machis.
Granada, in their first semi-final since 1969, then took a 2-1 overall lead thanks to a goal from German Sanchez after 76 minutes, again with Machis the provider from a corner.
However, their hopes of making the final were dashed when left-back Berchiche scored with a low drive in the 81st minute after being set up by Mikel Vesga to secure his team a spot in the April 18 finale in Seville.
With 23 titles, Athletic Bilbao are second on the all-time list of Copa del Rey winners behind Barcelona who have triumphed 30 times.