We propose an efficient algorithm for matching two correlated Erdős--Rényi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- α+o(1)}$ for a constant $α\in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).