The onset time of a coronal mass ejection (CME) is usually extrapolated from its speed in a coronagraph using the constant speed method. In this study, we present a new empirical method for extrapolating the onset times of flare-associated CMEs. For this we reexamined seven well-observed CME-flare events whose initial eruption speeds are estimated to be about a few tens of kilometers per second from LASCO C1 coronagraph or loop displacements seen in SOHO/EIT and TRACE images. In this study, the CME onset time is assumed to be the earliest observable time of eruption from these images. We estimated the onset time differences between the CMEs and the associated flares observed in soft X rays. We then compared them with the onset time differences estimated by the constant speed extrapolation method from LASCO C2 positions. It is noted that there is a certain quadratic relation between the CME speed first observed in the LASCO C2 field of view and the observed CME travel time from near the solar surface to the C2 field of view in units of the travel time estimated by the constant speed method. This empirical relation thus enables us to determine the onset times of CMEs without any assumption in CME kinematics. We have applied the new empirical method to 91 flare-associated CMEs that were accompanied by type II bursts. The onset time difference for these events as well as the well-observed events shows that in most cases (about 80%) the CME initiation precedes the onset of the associated flare. This result provides an important clue to the mechanism and relation of CMEs and flares.