The present invention relates to a wireless access system supporting millimeter waves (mmWave) and provides methods for performing fast fallback so as to avoid link disconnections and devices for supporting same. The method whereby a legacy base station supports the fast fallback of a mmWave terminal in a wireless access system supporting millimeter waves (mmWave), according to one embodiment of the present invention, may comprise the steps of: receiving, from the mmWave terminal, a first fallback request message for requesting fast fallback; operating a fallback timer for determining whether or not to perform fallback after receiving the first fallback request message; and determining whether or not a second fallback request message for requesting fast fallback has been received from a mmWave base station within the fallback timer.