Research on conjugated radiation-conduction (CRC) heat transfer in participating media is of vital scientific and engineering significance due to its extensive applications. Appropriate and practical numerical methods are essential to forecast the temperature distributions during the CRC heat-transfer processes. Here, we established a unified discontinuous Galerkin finite-element (DGFE) framework for solving transient CRC heat-transfer problems in participating media. To overcome the mismatch between the second-order derivative in the energy balance equation (EBE) and the DGFE solution domain, we rewrite the second-order EBE as two first-order equations and then solve both the radiative transfer equation (RTE) and the EBE in the same solution domain, resulting in the unified framework. Comparisons between the DGFE solutions with published data confirm the accuracy of the present framework for transient CRC heat transfer in one- and two-dimensional media. The proposed framework is further extended to CRC heat transfer in two-dimensional anisotropic scattering media. Results indicate that the present DGFE can precisely capture the temperature distribution at high computational efficiency, paving the way for a benchmark numerical tool for CRC heat-transfer problems.