Spatial and spatio-temporal cluster detection are important tools in public health and many other areas of application. Cluster detection can be approached as a multiple testing problem, typically using a space and time scan statistic. We recast the spatial and spatio-temporal cluster detection problem in a high-dimensional data analytical framework with Poisson or quasi-Poisson regression with the Lasso penalty. We adopt a fast and computationally-efficient method using a novel sparse matrix representation of the effects of potential clusters. The number of clusters and tuning parameters are selected based on (quasi-)information criteria. We evaluate the performance of our proposed method including the false positive detection rate and power using a simulation study. Application of the method is illustrated using breast cancer incidence data from three prefectures in Japan.