A fluid dispenser device having a reservoir (10) containing fluid; a dispenser head (20) provided with a dispenser orifice (21); and a piston (45; 70) for dispensing at least a portion of the fluid through the dispenser orifice (21). The reservoir (10) is closed in leaktight manner by a closure element (25) before actuation. The device includes a passage (40) for connecting the reservoir (10) to the dispenser orifice (21) during actuation, the passage (40) including a hollow sleeve (41) that is stationary relative to the dispenser head (20), for sliding around the closure element (25) during actuation, so as to open the passage (40).