Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.