It is known that in cases of locally connected topological spaces or compact Hausdorff spaces the components and quasicomponents coincide. Both claims can be proved using the fact that if a quasicomponent is open, then it is connected.
Does the opposite of the last claim hold too, i.e., if the component is open, does it imply that it must coincide with a quasicomponent?
Yes. If a component $C$ is open, then it is clopen. It cannot contain a non-empty strictly smaller clopen subset, thus it is the intersection of all clopen subsets containing a fixed point $x \in C$.