Clusters of galaxies have been found to host Mpc-scale diffuse, non-thermal radio emission in the form of central radio halos and peripheral relics. Turbulence and shock-related processes in the intra-cluster medium are generally considered responsible for the emission, though details of these processes are still not clear. The low surface brightness makes detection of the emission a challenge, but with recent surveys with high-sensitivity radio telescopes we are beginning to build large samples of these sources. The Evolutionary Map of the Universe (EMU) is a Southern Sky survey being performed by the Australian SKA Pathfinder (ASKAP) over the next few years and is well-suited to detect and characterise such emission. To assess prospects of the full survey, we have performed a pilot search of diffuse sources in 71 clusters from the Planck Sunyaev–Zeldovich (SZ) cluster catalogue (PSZ2) found in archival ASKAP observations. After re-imaging the archival data and performing both (u, v)-plane and image-plane angular scale filtering, we detect 21 radio halos (12 for the first time, excluding an additional six candidates), 11 relics (in seven clusters, and six for the first time, excluding a further five candidate relics), along with 12 other, unclassified diffuse radio sources. From these detections, we predict the full EMU survey will uncover up to ≈ 254 radio halos and ≈ 85 radio relics in the 858 PSZ2 clusters that will be covered by EMU. The percentage of clusters found to host diffuse emission in this work is similar to the number reported in recent cluster surveys with the LOw Frequency ARray (LOFAR) Two-metre Sky Survey (Botteon, et al. 2022a, A&A, 660, A78), suggesting EMU will complement similar searches being performed in the Northern Sky and provide us with statistically significant samples of halos and relics at the completion of the full survey. This work presents the first step towards large samples of the diffuse radio sources in Southern Sky clusters with ASKAP and eventually the SKA.