Taylor resigned for personal reasons this week, a few days after the Wallabies conceded seven tries in a record 48-17 loss to Argentina and ahead of back-to-back home tests against the world champion Springboks.
Fisher, currently in charge of the forwards at the Brumbies, is one of the most respected coaches in Australian rugby for his work at the Canberra-based team over the last two decades.
"We're grateful Laurie has been able to join us at such short notice and would also like to thank the Brumbies for their support too," said head coach Dave Rennie.