Femtosecond vibrational wave-packet dynamics in a cyanine molecule is observed to be strongly dependent on the chirp direction of the excitation pulse. The slow-decay component associated with an oscillatory structure, which corresponds to the excited-state lifetime and a vibrational mode of 160 +/- 10 cm(-1), respectively, is measured by the femtosecond time-resolved transmission spectroscopy. The excited-state population is substantially decreased and enhanced in the cases of negatively chirped (NC) and positively chirped (PC) excitations, respectively. A quantum mechanical calculation by means of the split operator scheme is performed to reproduce the wave-packet propagation after the chirped pulse excitation. The calculation shows that the spatial distribution of the wave packet for the NC case is narrower than that for the PC case during the excitation, and that the overlap integral between the excited- and ground-state wave packets determines the efficiency of the population dumping.