Despite significant achievements in characterizing the properties of Sr$_2$RuO$_4$ over the last three decades, the precise nature of its electronic ground state is still unresolved. In this work, we provide a missing piece of the puzzle by uncovering evidence of electronic nematic order in the normal state of Sr$_2$RuO$_4$, revealed by ultrafast time-resolved optical dichroism measurements of uniaxially strained thin films. This nematic order, whose domains are aligned by the strain, spontaneously breaks the four-fold rotational symmetry of the crystal. The temperature dependence of the dichroism resembles an Ising-like order parameter, and optical pumping induces a coherent oscillation of its amplitude mode. A microscopic model of intra-unit-cell nematic order is presented, highlighting the importance of Coulomb repulsion between neighboring oxygen $p$-orbitals. The existence of electronic nematic order in the normal state of Sr$_2$RuO$_4$ may have consequences for the form and mechanism of superconductivity in this material.