We perform a detailed analysis of the latest HARPS and Keck radial velocity data for the planet-hosting red dwarf GJ 581, which attracted a lot of attention in recent time. We show that these data contain important correlated noise component (`red noise') with the correlation time-scale of the order of 10 d. This red noise imposes a lot of misleading effects while we work in the traditional white-noise model. To eliminate these misleading effects, we propose a maximum-likelihood algorithm equipped by an extended model of the noise structure. We treat the red noise as a Gaussian random process with an exponentially decaying correlation function. Using this method we prove that (i) planets b and c do exist in this system, since they can be independently detected in the HARPS and Keck data, and regardless of the assumed noise models; (ii) planet e can also be confirmed independently by both the data sets, although to reveal it in the Keck data it is mandatory to take the red noise into account; (iii) the recentl