A criterion is given for showing that certain one-relator groups are residually finite. This is applied to a one-relator group with torsion $G = \langle a_1, \ldots, a_r \mid W^n\rangle$. It is shown that $G$ is residually finite provided that $W$ is outside the commutator subgroup and $n$ is sufficiently large. An important ingredient in the proof is a criterion which implies that a subgroup of a group is malnormal. A graded small-cancellation criterion is developed which detects whether a map $A \rightarrow B$ between graphs induces a $\pi_1$-injection, and whether $\pi_1 A$ maps to a malnormal subgroup of $\pi_1 B$.