The pth moment exponential synchronization (PMES) problem for time-varying complex dynamical networks (CDNs) with multiple time-varying delays (TCDNMTDs) by aperiodically intermittent control is studied in this paper. Compared with related work, the coupling structure of CDNs is time varying. Also, internal time-varying delays and coupling time-varying delays are taken into consideration and they are different from each other. It is worth emphasizing that the intermittent control scheme proposed here is nonperiodical. Some criteria are derived to make TCDNMTDs achieve PMES by using the Lyapunov method and graph theory. Additionally, we also consider the synchronization of second-order Kuramoto oscillators and coupled Chua's circuits and some numerical simulations are presented to verify the validity of our results.