We study the effect of Coulomb drag between two closely positioned graphene monolayers, assuming that transport properties of the sample are dominated by disorder. This assumption allows us to develop a perturbation theory in ${\ensuremath{\alpha}}^{2}T\ensuremath{\tau}\mathrm{min}(1,T/{\ensuremath{\mu}}_{1(2)})\ensuremath{\ll}1$ (here, $\ensuremath{\alpha}={e}^{2}/v$ is the strength of the Coulomb interaction, $T$ is temperature, ${\ensuremath{\mu}}_{1(2)}$ is the chemical potential of the two layers, and $\ensuremath{\tau}$ is the mean-free time). Our theory applies for arbitrary values of ${\ensuremath{\mu}}_{1(2)}$, $T$, and the interlayer separation $d$, although we focus on the experimentally relevant situation of low temperatures $T<v/d$. We find that the drag coefficient ${\ensuremath{\rho}}_{D}$ is a nonmonotonous function of ${\ensuremath{\mu}}_{1(2)}$ and $T$ in qualitative agreement with experiment [S. Kim et al., Phys. Rev. B 83, 161401(R) (2011)]. Precisely at the Dirac point, drag vanishes due to electron-hole symmetry. For very large values of chemical potential ${\ensuremath{\mu}}_{1(2)}\ensuremath{\gg}v/(\ensuremath{\alpha}d)$, we recover the standard Fermi-liquid result ${\ensuremath{\rho}}_{D}^{\text{FL}}\ensuremath{\propto}{T}^{2}{({n}_{1}{n}_{2})}^{\ensuremath{-}3/2}{d}^{\ensuremath{-}4}$ (where ${n}_{i}$ is the carrier density in the two layers). At intermediate values of the chemical potential, the drag coefficient exhibits a maximum. The decrease of the drag coefficient as a function of ${\ensuremath{\mu}}_{1(2)}$ (or the carrier densities) from its maximum value is characterized by a crossover from a logarithmic behavior ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{T}^{2}{({n}_{1}{n}_{2})}^{\ensuremath{-}1/2}\mathrm{ln}{n}_{1}$ to the Fermi-liquid result. Our results do not depend on the microscopic model of impurity scattering. The crossover occurs in a wide range of densities, where ${\ensuremath{\rho}}_{D}$ can not be described by a power law. On the contrary, the increase of the drag from the Dirac point (for ${\ensuremath{\mu}}_{1(2)}\ensuremath{\ll}v/d$) is described by the universal function of ${\ensuremath{\mu}}_{1(2)}$ measured in the units of $T$; if both layers are close to the Dirac point ${\ensuremath{\mu}}_{1(2)}\ensuremath{\ll}T$, then ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}/{T}^{2}$; in the opposite limit of low temperature, ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{T}^{2}/({\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2})$ and in the mixed case, ${\ensuremath{\mu}}_{1}\ensuremath{\ll}T\ensuremath{\ll}{\ensuremath{\mu}}_{2}$, we find ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{\ensuremath{\mu}}_{1}/{\ensuremath{\mu}}_{2}$.