Eigenvector centrality mapping (ECM) has recently emerged as a measure to spatially characterize connectivity in functional brain imaging by attributing network properties to voxels. The main obstacle for widespread use of ECM in functional magnetic resonance imaging (fMRI) is the cost of computing and storing the connectivity matrix. This article presents fast ECM (fECM), an efficient algorithm to estimate voxel-wise eigenvector centralities from fMRI time series. Instead of explicitly storing the connectivity matrix, fECM computes matrix-vector products directly from the data, achieving high accelerations for computing voxel-wise centralities in fMRI at standard resolutions for multivariate analyses, and enabling high-resolution analyses performed on standard hardware. We demonstrate the validity of fECM at cluster and voxel levels, using synthetic and in vivo data. Results from synthetic data are compared to the theoretical gold standard, and local centrality changes in fMRI data are measured after experimental intervention. A simple scheme is presented to generate time series with prescribed covariances that represent a connectivity matrix. These time series are used to construct a 4D dataset whose volumes consist of separate regions with known intra- and inter-regional connectivities. The fECM method is tested and validated on these synthetic data. Resting-state fMRI data acquired after real-versus-sham repetitive transcranial magnetic stimulation show fECM connectivity changes in resting-state network regions. A comparison of analyses with and without accounting for motion parameters demonstrates a moderate effect of these parameters on the centrality estimates. Its computational speed and statistical sensitivity make fECM a good candidate for connectivity analyses of multimodality and high-resolution functional neuroimaging data.