A fluid dispenser device including a body (10); a reservoir (20) containing fluid; a dispenser member (30), such as a pump or a valve, for dispensing the fluid contained in the reservoir (20) through a dispenser orifice (45); and a cap (50) that is adapted to cover said dispenser orifice (45) in a closed position. The cap (50) is pivotally mounted on the body (10) to turn about an axis (A) between the closed position and an actuation position, the cap (50) including an actuator mechanism (55; 56, 57) adapted to actuate the dispenser member (30) directly, while the cap (50) is turning towards its actuation position.