Let $α\in[0,1)$, and let $G$ be a graph of even order $n$ with $n\geq f(α)$, where $f(α)=10$ for $0\leq α\leq1/2$, $f(α)=14$ for $1/2<α\leq 2/3$ and $f(α)=5/(1-α)$ for $2/3<α<1$. In this paper, it is shown that if the $A_α$-spectral radius of $G$ is not less than the largest root of $x^3 - ((α+ 1)n +α-4)x^2 + (αn^2 + (α^2 - 2α- 1)n - 2α+1)x -α^2n^2 + (5α^2 - 3α+ 2)n - 10α^2 + 15α- 8=0$ then $G$ has a perfect matching unless $G=K_1\nabla(K_{n-3}\cup 2K_1)$. This generalizes a result of S. O [Spectral radius and matchings in graphs, Linear Algebra Appl. 614 (2021) 316--324], which gives a sufficient condition for the existence of a perfect matching in a graph in terms of the adjacency spectral radius.