We investigate theoretically and experimentally the singlet-triplet Kondo effect induced by a magnetic field in a molecular junction. Temperature-dependent conductance, G(T), is calculated by the numerical renormalization group, showing a strong imprint of the relevant low-energy scales, such as the Kondo temperature, exchange, and singlet-triplet splitting. We demonstrate the stability of the singlet-triplet Kondo effect against weak spin anisotropy, modeled by an anisotropic exchange. Moderate spin anisotropy manifests itself by lowering the Kondo plateaus, causing the G(T) to deviate from a standard temperature dependence, expected for a spin-half Kondo effect. We propose this scenario as an explanation for anomalous G(T), measured in an organic diradical molecule coupled to gold contacts. We uncover certain new aspects of the singlet-triplet Kondo effect, such as coexistence of spin-polarization on the molecule with Kondo screening and nonperturbative parametric dependence of an effective magnetic field induced by the leads.