I have posted an answer here for the case of an affine scheme, but I got stuck when I tried to generalize the argument to schemes.
My thoughts
Consider a point $p$ in the scheme, its closure in the scheme is an irreducible closed subset. This gives us one direction of the correspondence.
Let $C$ be an irreducible closed subset of the scheme, pick an affine neighborhood $U$ that intersects nontrivially with $C$. Then the intersection is a closed subset of $U$ which decomposes into finite union of irreducible closed subsets of $U$ by Noetherian property of $U$. This is where I got stuck, and don't know how to proceed from here.
Hint: First the closure of a point is always irreducible, there's no need to pass through an affine neighborhood for that.
For the other direction if $C$ is an irreducible closed set and $U$ is open then show that $C \cap U$ is an irreducible closed subset of $U$.
Now $U \cap C$ has a unique generic point when $U$ is affine. Show that if $V$ is also affine and $V \cap C$ is nonempty then the generic point of $U \cap C$ lies in $V$ and hence equals the unique generic point of $V \cap C$.
Finally use the fact that affine opens form a base for the topology to show that the common generic point for all nonempty $U \cap C$ is a generic point for $C$, necessarily a unique one.