It is likely not this one, but I thought I would throw it in anyways because it is the first thing I thought of when reading the question.
It might be Empire of the East.
This book starts as a typical fantasy novel with some strange magic. There are instances of Technology in the world, but magic is predominant. It is not initially explained what happened and the main characters only know of myths and legends which are not very informative.
By the end of the book, we find out what happened. It turns out that the reason for the fall of technology was a large supercomputer named ARDNEH. It was a nuclear fail-safe which was supposed to activate in the case of a nuclear attack to protect everyone. It did so by invoking a "change" which we are led to believe manipulated the laws of physics in such a way as to negate nuclear fusion bombs. When WWIII happened, ARDNEH activated "the change". The unexpected result was that it also opened the door to magic and made some technology no longer viable due to the new physical laws.
The characters do come across a large furry beast, but don't have to fight it. It is actually an evolved dog and is named Draffut who had gained powers and intelligence from a magical healing pond (originally some technological feat as well). They do, however, have to fight demons. One demon, Orcus, was the result of a nuclear blast going off at the exact time of "the change". The blast froze mid explosion and became pure power, that (like Draffut) gained sentience over time and turned into a powerful malevolent being. Likewise, ARDNEH became sentient as well and was able to keep itself running though the years.
At the climax of the book, ARDNEH is under attack by Orcus and just when it is about to destroy ARDNEH, "the change" is reversed. This causes Orcus to finally complete his nuclear explosion, killing him and ARDNEH in the process. However, there is still magic in the world after and there is a subsequent book, ARDNEH's Sword, which leads into Saberhagen's Swords series.