Let $u,v$ be two analytic functions on the open unit disk $\mathbb{D}$ in the complex plane, $\phi$ an analytic self-map of $\mathbb{D}$, and $m,n$ nonnegative integers such that $m<n$. In this paper, we consider the generalized Stević-Sharma type operator $T_{u,v,\phi }^{m,n}f(z)=u(z)f^{(m)}(\phi (z))+v(z)f^{(n)}(\phi (z))$ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness.