Compressive random access (CRA) is a random access scheme that has been considered for massive machine-type communication (MTC) with non-orthogonal spreading sequences, where the notion of compressive sensing (CS) is used for low-complexity detectors by exploiting the sparse device activity. In this paper, we study the application of power-domain non-orthogonal multiple access (NOMA) to CRA in order to improve the performance of CRA (or increase the number of devices to be supported). We consider Gaussian spreading sequences and derive design criteria through a large-system analysis to determine the power levels for power-domain NOMA. From simulation results, we can confirm that the number of incorrectly detected device activity can be reduced by applying NOMA to CRA.