Rescues by eagles in two Tolkien books have been described as examples of this dramatic device (4,2,7)

I believe the answer is:
deus ex machina
(Other definitions for deus ex machina that I've seen before include "unlikely device to resolve difficulty" , "Contrived agent who unexpectedly solves the insoluble" , "Late delivery" , "Unexpected problem solver" , "rather improbable device" .)