CGI has been used in various forms for years. It started to replace animations as special effects in the movies in the early 1970's. It was first used as part of a feature length animation in Disney's The Black Cauldron in 1985.
Computer generated animation took a leap forward with Pixar's use of shadow in the short film, Luxo Jr*. A lot of the development of CGI effects were developed with it's use in feature films. As the ability to provide more memory and processing power in a small space grew so did the improvements in CGI. This, in turn, led to the first complete CGI feature length animation, Toy Story*, by Pixar in 1995.
Most animations produced now have some form of computer software to help generate them, even those that have all of the artwork done by hand. Some people have a preference for one or the other but without the processing power of computers the current resurgence of animation would not have taken place. Interestingly, Disney's 2012 short film, Paperman*, was an attempt to seamlessly blend CGI and handmade artwork.