This paper completes the determination of non-Cayley numbers of the form \(2pq\) (\(p\) and \(q\) are distinct odd primes), where a non-Cayley number is a natural number \(n\) for which there exists a vertex-transitive graph on \(n\) vertices which is not a Cayley graph. For vertex-transitive graphs admitting an imprimitive subgroup, this question was answered by \textit{A. A. Miller} and \textit{C. E. Praeger} [J. Algebr. Comb. 3, No. 1, 77-111 (1994; Zbl 0794.05046)]. Here the primitive permutation groups of degree \(2pq\) are classified using the classification of finite simple groups. Then each of these groups is tested to determine whether any non-Cayley graph admits it as a vertex-primitive subgroup of automorphisms while admitting no imprimitive subgroups, resulting in the following classification: \(n=2pq\) is a non-Cayley number if and only if \(n=154\) or \(n=266\), or \(p\) or \(q\) are congruent 1 modulo 4, or \(p\) is congruent 1 modulo \(q^2\), or \(p=4q-1\).