Dynamic contrast-enhanced MRI (DCE-MRI) is an imaging protocol where MRI scans are acquired repetitively throughout the injection of a contrast agent. The analysis of dynamic scans is widely used for the detection and quantification of blood-brain barrier (BBB) permeability. Extraction of the pharmacokinetic (PK) parameters from the DCE-MRI concentration curves allows quantitative assessment of the integrity of the BBB functionality. However, curve fitting required for the analysis of DCE-MRI data is error-prone as the dynamic scans are subject to non-white, spatially-dependent and anisotropic noise. We present a novel spatio-temporal framework based on Deep Neural Networks (DNNs) to address the DCE-MRI denoising challenges. This is accomplished by an ensemble of expert DNNs constructed as deep autoencoders, where each is trained on a specific subset of the input space to accommodate different noise characteristics and curve prototypes. Spatial dependencies of the PK dynamics are captured by incorporating the curves of neighboring voxels in the entire process. The most likely reconstructed curves are then chosen using a classifier DNN followed by a quadratic programming optimization. As clean signals (ground-truth) for training are not available, a fully automatic model for generating realistic training sets with complex nonlinear dynamics is introduced. The proposed approach has been successfully applied to full and even temporally down-sampled DCE-MRI sequences, from two different databases, of stroke and brain tumor patients, and is shown to favorably compare to state-of-the-art denoising methods.