We describe the stable existence of quasi-one-dimensional solutions of the two-dimensional cubic-quintic complex Ginzburg-Landau equation for a large range of the bifurcation parameter. By quasi-one-dimensional (quasi-1D) in the present context, we mean solutions of fixed shape in one spatial dimension that are simultaneously fully extended and space filling in a second direction. This class of stable solutions arises for parameter values for which simultaneously other classes of solutions are at least locally stable: the zero solution, 2D fixed shape dissipative solitons, or 2D azimuthally symmetric or asymmetric exploding dissipative solitons. We show that quasi-1D solutions can form stable compound states with 2D stationary dissipative solitons or with azimuthally symmetric exploding dissipative solitons. In addition, we find stable breathing quasi-1D solutions near the transition to collapse. The analogy of several features of the work presented here to recent experimental results on convection by Miranda and Burguete [Phys. Rev. E 78, 046305 (2008); Phys. Rev. E 79, 046201 (2009)] is elucidated.