Solution: Starting from point $P$ at $(1,0)$, moving counterclockwise along the unit circle for an arc length of $\dfrac {2\pi}{3}$ reaches point $Q$,
so $\angle QOx= \dfrac {2\pi}{3}$,
thus $Q(\cos \dfrac {2\pi}{3},\sin \dfrac {2\pi}{3})$,
which means the coordinates of point $Q$ are: $\left(- \dfrac {1}{2}, \dfrac { \sqrt {3}}{2}\right)$.
Therefore, the answer is: $\boxed{A}$.

The problem involves deducing the angle $\angle QOx$ and then finding the coordinates of point $Q$. This question tests computational skills through the rotation of the terminal side of an angle, focusing on the direction of rotation, and is considered a basic question.