We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $μ>0$, there exist $α>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $δ(G)\geq \left(1-\frac{1}{r}+μ\right)n$ and $α(G)\leq αn$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.