We prove, for certain pairs G,G' of finite groups of Lie type, that the $p$-fusion systems F_p(G) and F_p(G') are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie "type" H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result.