We recently found extended CO(2-1) emission from cold molecular gas embedded in the network of Ha filaments surrounding the galaxy NGC 1275 (Salome et al. 2006). We now present CO(2-1) interferometer maps of the eastern filaments, at high spatial and spectral resolutions. The cold molecular gas is detected by the Plateau de Bure Interferometer along the eastern filaments over an extent of 15'', or with a projected length of 5 kpc. In our 2.5'' beam, the main CO filament is mostly unresolved along its minor axis. The multiple peaks along the CO filaments and the low values of the observed CO brightness temperatures imply further unresolved structures that may be giant molecular clouds. These clouds have very narrow line-width emission lines (similar to 30 km s(-1)). The CO emission is optically thick. It very likely traces cold clouds bound under their own self-gravity that may be falling back in the gravitational potential well of the galaxy. Such a picture would agree with current models of "positive feedback" in which some of the hot gas around NGC 1275 (a) is trapped by buoyantly rising bubbles inflated by the energy input of the 3C 84 AGN, (b) subsequently cools effciently at a larger radius around the edges of the hot bubbles, and (c) then falls back in self-gravitating clouds of molecular gas toward the center of the galaxy.