(This answers the previous version of the question, for components but not quasicomponents).
I think, it us true in every topological space: if $X$ is a topological space with connected components $P$ and $Q_j$: $X=P\sqcup (\sqcup_j Q_j)$, and you remove a set $C$ from a connected component $P$ so that it becomes disconnected: $P\setminus C=\sqcup A_i$, where $A_i$ are the connected components of $P\setminus C$, then the connected components of $X\setminus C$ are $A_i$'s and $Q_j$'s. The reason is that the connected components are the inclusion-maximal connected subsets, and each connected subset of $X\setminus C$ lies inside a single connected component of $X$.