Yes, there are certain ways that a species of animals can be saved after they have reached functional extinction. However, in most cases they don’t succeed in getting out of extinction once they have been termed as functionally extinct. Usually, animals that are in smaller populations at the time of extinction are considered functionally extinct.
However, there are ways in which these animals that are in medium or larger populations may either not reach functional extinction or they may be able to get out of being considered functionally extinct. Like any kind of extinction, these animals would have to be in a sanctuary for those animals where they could easily survive and easily live in a habitat that is close to their environment in the wild.
In theory, it is possible to bring back a species from functional extinction. In fact, the Pyrenean ibex, a type of wild goat, was brought back from extinction in 2003. Unfortunately, the Pyrenean ibex died out again, shortly thereafter. Scientists also tried to artificially inseminate the last two West African Black Rhinoceros females with sperm from a male that had passed away previously.
This attempt didn't work either due to the females' age and stress levels. It is more likely to see animals come out of functional extinction before their numbers get too close to numerical extinction, but it still doesn't mean this will prevent the extinction of the species.