A fluid flow control valve includes a valve body with a fluid flow lumen therethrough and a valve member located within the fluid flow lumen. At least one of the valve body and valve member is resilient and respectively sized for movement between a normally closed position in which the valve member and a fluid lumen surface are in sufficient annular contact to block fluid flow through the fluid flow lumen and an open position in which at least one of the valve body and valve member flexes to a spaced apart position that allows fluid flow between them. Downstream fluid pressure may contact the valve body to enhance sealing. A fluid absorbent member may also be employed with the valve body. Flow control devices, such as medical fluid injection devices, and methods may employ this valve.