We recently found extended CO(2-1) emission from cold molecular gas embedded in the network of Hα 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...