While extant is (roughly) the opposite of extinct (the state of [still] existing vs. the state of not existing), I would contend that there isn't an analogous opposite to extinction:
1a. The act of extinguishing: The extinction of the fire took several hours.
1b. The condition of being extinguished: mourned the extinction of her dreams.
(Both from American Heritage Dictionary)
1. The act of making extinct or the state of being extinct
(From Collins English Dictionary)
For the first of these senses (the act of extinguishing/making extinct) then the opposite is the act of bringing into existence. At least for species, and depending on one's beliefs, you might have:
The formation of new biological species through the process of evolution.
- The divine act by which, according to various religious and philosophical traditions, the world was brought into existence.
(Creation, in a non-religious sense could be used for the coming-into-existence of most things, but probably wouldn't be used for species, because of the religious overtones).
For the second sense (the state of being extinct), then survival (as in lbf's answer) is probably the closest opposite, although (to me, at least) it's not perfect: extinction is a definite thing that has happened; survival is the ongoing state of not yet having been made extinct.