Analysis of neutron diffraction, dc magnetization, ac magnetic susceptibility, heat capacity, and electrical resistivity for DyRuAsO in an applied magnetic field are presented at temperatures near and below those at which the structural distortion (T_S = 25 K) and subsequent magnetic ordering (T_N = 10.5 K) take place. Powder neutron diffraction is used to determine the antiferromagnetic order of Dy moments of magnitude 7.6(1) mu_B in the absence of a magnetic field, and demonstrate the reorientation of the moments into a ferromagnetic configuration upon application of a magnetic field. Dy magnetism is identified as the driving force for the structural distortion. The magnetic structure of analogous TbRuAsO is also reported. Competition between the two magnetically ordered states in DyRuAsO is found to produce unusual physical properties in applied magnetic fields at low temperature. An additional phase transition near T* = 3 K is observed in heat capacity and other properties in fields greater than about 3 T. Magnetic fields of this magnitude also induce spin-glass-like behavior including thermal and magnetic hysteresis, divergence of zero-field-cooled and field-cooled magnetization, frequency dependent anomalies in ac magnetic susceptibility, and slow relaxation of the magnetization. This is remarkable since DyRuAsO is a stoichiometric material with no disorder detected by neutron diffraction, and suggests analogies with spin-ice compounds and related materials with strong geometric frustration.