We do not have a wealth of information on the subject doctrinally, but the Prophet Joseph Smith did teach that animals have spirits and that they do exist in the spirit world as well they will be resurrected and gain immortality.  
 
The Prophet Joseph Smith also taught it was a sin to needlessly kill animals, that those who shed the blood of animals without need for food or clothing are committing sin. Now I cannot say if putting an animal to sleep is right or wrong scripturally because there is not scriptures that cover that topic.  
 
Personally I see no real sin in relieving the pain of an animal that is at the end of it's mortal existence. I do not believe that the Church as a doctrinal stance on this issue either. Even if it is a transgression to put an animal to sleep we still can rely upon the atonement of Jesus Christ. I do believe though that this is murder.

I believe that you did the right thing! If it was suffering then it needed to be free from the pain and in a better place. I think it would be wrong if you didn't and just watch it suffer.