In 2009 January, the 2.1-s anomalous X-ray pulsar 1E 1547.0−5408 evoked intense burst activity. A follow-up Suzaku observation on January 28 recorded enhanced persistent emission in both soft and hard X-rays. Through a reanalysis of the same Suzaku data, 18 short bursts were identified in the X-ray events recorded by the Hard X-ray Detector (HXD) and the X-ray Imaging Spectrometer (XIS). Their spectral peaks appear in the HXD–PIN band, and their 10–70 keV X-ray fluences range from ∼2 × 10−9 to 10−7 erg cm−2. Thus, the 18 events define a significantly weaker burst sample than has ever been obtained previously, ∼10−8–10−4 erg cm−2. In the ∼0.8 to ∼300 keV band, the spectra of the three brightest bursts can be represented successfully by a two-blackbody model, or a few alternative models. A spectrum that is constructed by stacking 13 weaker short bursts with fluences in the range (0.2–2) × 10−8 erg s−1 is less curved, and its ratio to the persistent emission spectrum becomes constant at ∼170 above ∼8 keV. As a result, the two-blackbody model was able to reproduce the stacked weaker-burst spectrum only after adding a power-law model, for which the photon index is fixed at 1.54 as measured by the persistent spectrum. These results imply that there is a possibility that the spectrum composition that employs an optically thick component and a hard power-law component can describe the wide-band spectra of both the persistent and weak-burst emissions, despite the fact that their fluxes differ by two orders of magnitude. Based on the spectral similarity, we discuss a possible connection between the unresolved short bursts and the persistent emission.