Examples of humans intentionally introducing non-native species where everything went well?
The best examples are for when non-native pest species arrive, and then a search for their natural enemies in their native countries is undertaken to bring them back and control the invaders. This is called "biocontrol", and there are many examples of it going well, under carefully controlled conditions. https://biocontrol.entomology.cornell.edu/success.php 
However, you are right to note that it is very difficult to predict.