In this study, the problem of output consensus control for high-order continuous-time linear multi-agent systems with interval time-varying delays is investigated. The observability decomposition technique is employed to design the output consensus protocols which are the collections of delayed output information from neighbouring agents. On the basis of invertible transformations, output consensus for the concerned dynamical agents is transformed into the problem of asymptotical stability analysis for some lower dimensional subsystems. By introducing the prescribed convergence rate scalar into the constructed delay-dependent Lyapunov functionals, the framework for output consensus analysis is theoretically derived. Benefitting from the method of slack matrix variables, sufficient conditions are obtained in terms of linear matrix inequalities to design the protocol gain matrices which can guarantee the property of output consensus. Moreover, the output consensus function determined merely by the initial states of dynamic agents and consensus protocol is presented without the influences of time-varying delays. Numerical examples are exploited to illustrate the effectiveness of the derived results.