Argentina refused to yield and Nicolas Sanchez's flawless boot - he kicked five penalties in all - saw them trail by seven points with 26 minutes to go.
But Ashley-Cooper completed a hat-trick of tries late on to secure victory.
Roared on by a passionate support that included football World Cup winner Diego Maradona, Argentina had chances in the second half as they took control up front and utilised their explosive runners.
But they lacked the composure to exploit several possible overlaps and Australia will now face neighbours New Zealand in the final.
The two great Antipodean rivals have never met before in a World Cup final, and a tournament that has been blessed with so many epic contests may yet have one more.