A version of OP's claim (with an additional hypothesis) is mentioned in a footnote on the first page of
- Borel, A. Injective endomorphisms of algebraic varieties. Arch. Math 20, 531–537 (1969). https://doi.org/10.1007/BF01899460,
where he says that M. Raynaud proved the following:
If $V$ is an algebraic variety, $f: V\to V$ has finite fibers, and is an immersion on a dense open subset of $V$, then $f$ is an isomorphism.