We find two involutions on partitions that lead to partition identities for Ramanujan's third-order mock theta functions ϕ(−q) and ψ(−q). We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third-order mock theta functions are consequences of these identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan's identities.