This paper investigates synchronization dynamics of a complex delayed dynamical network as well as the effects of time-varying delay. Following the approach via Razumikhin theorem, simple delay-dependent synchronization criteria are derived in terms of linear matrix inequalities, which can be verified via the interior-point algorithm. The proposed criteria can deal with a fast time-varying delay in coupling term and enabled removing the restriction on the derivative of the time-varying delay. The effectiveness of the proposed synchronization scheme and the theoretical results are illustrated by a numerical example.