We found that models of evolving random networks exhibit dynamic scaling similar to scaling of growing surfaces. It is demonstrated by numerical simulations of two variants of the model in which nodes are added as well as removed (Phys. Rev. Lett., 83 (1999) 5587). The averaged size and connectivity of the network increase as power laws in early times but later saturate. Saturated values and times of saturation change with paramaters controlling the local evolution of the network topology. Both saturated values and times of saturation obey also power law dependences on controlling parameters. Scaling exponents are calculated and universal features are discussed.