The answer is "no" without some further hypothesis on $X$ and $Y$. For instance, let $Y$ be $\text{Spec}(\mathbb{Z}_{5\mathbb{Z}})$, the localization of $\mathbb{Z}$ at the prime ideal $5\mathbb{Z}$, let $X$ be $\text{Spec}(\mathbb{Z}_{5\mathbb{Z}}[x]/\langle x^2 + 1 \rangle)$, and let $U$ be the distinguished open where $5$ is invertible.
However, for finite type schemes over a field, this does follow from Zariski's Main Theorem.
Edit. I did not see jmc's answer below when I posted this answer. This answer is basically the same as jmc's answer. It is important to note, even for finite type schemes over a field, being injective on points does not necessarily imply the morphism is birational, e.g., Frobenius morphisms.