An elastic support device (70) has a vibration reduction member (35) for supporting a vibration transmission medium (34). The vibration transmission medium (34) is connected to the vibration reduction member (35) by a connection member (42). The vibration reduction member (35) includes a sleeve (43) made from a rigid body, through which sleeve the connection member (42) is inserted, and an elastic member (45) surrounding an outer periphery of the sleeve (43). The elastic member (45) includes a tubular body (49) firmly fixed on an outer peripheral surface (44) of the sleeve (43) and an annular plate portion (50) projecting radially outward of the body (49). A fitting groove (58) of an outer peripheral edge portion (57) of the annular plate portion (50) is fitted on a peripheral edge (60) of an installation hole (59). A first and a second face (52, 53) of the annular plate portion (50) include recesses (54, 55), respectively. In the annular plate portion (50), a thin-walled portion (56) corresponding to a portion between bottoms of the recesses (54, 55) of the first and the second faces (52, 53) is formed.