A fluid-filled elastic mount including a first and a second support member, an elastic body for elastically connecting the two support members, and a closure member fixed to the second support member to form a fluid chamber therein. The closure member includes a flexible diaphragm and a metallic member secured to the outer periphery of the diaphragm. The metallic member is press-fitted in a large-diameter portion of the second support member remote from the first support member, and fixed by caulking to the large-diameter portion. At least one sealing rubber lip is pressed between the metallic member and the second support member to assure the fluid tightness of the fluid chamber. Further, a fluid discharge passage is formed radially outwardly of the sealing rubber lip(s), between the metallic member and the large-diameter portion of the second support member, for permitting the fluid to be discharged from the fluid chamber. Also disclosed is a method of producing such a fluid-filled elastic mount.