The effect of third-order dispersion on breathing localized solutions in the quintic complex Ginzburg-Landau (CGL) equation is investigated. It is found that even small third-order dispersion can cause dramatic changes in the behavior of the solutions, such as breathing solution asymmetrically and travelling slowly towards the right for the positive third-order dispersion. A little larger third-order dispersion causes the solution breathing only on one side and the other side keeping the soliton profile. For the negative dispersion, the same results can be obtained except for the change of the traveling direction. Otherwise, we analyzed the interaction of two breathing solitons and found a simple method to inhibit this interaction.