We present the first detection of CO emission lines in the Ha filaments at distances as far as 50 kpc from the centre of the galaxy NGC 1275. This gas is probably dense (>= 10(3) cm(-3)). However, it is not possible to accurately determine the density and the kinetic temperature of this relatively warm gas (T-kin similar to 20-500 K) with the current data alone. The amount of molecular gas in the filaments is large -10(9) M-circle dot ( assuming a Galactic N(H-2)/I-CO ratio). This is 10% of the total mass of molecular gas detected in this cD galaxy. This gas has large-scale velocities comparable to those seen in H alpha. The origin of the filaments is still unclear, but their formation is very likely linked to the AGN positive feedback that regulates the cooling of the surrounding X-ray-emitting gas as suggested by numerical simulations. We also present high-resolution spectra of the galaxy core. The spatial characteristics of the double-peaked profile suggest that the molecular web of filaments and streamers penetrates down to radii of less than 2 kpc from the central AGN and eventually feeds the galaxy nucleus. The mass of gas inside the very central region is similar to 10(9) M-circle dot, and is similar to the mass of molecular gas found in the filaments.