A product dispenser includes a housing having a backplate and a cover that, together, define a reservoir recess, and a refill unit received in the housing, the refill unit including a product reservoir positioned in the reservoir recess and holding product for dispensing, and a pump carried by the product reservoir and actuated to dispense the product. The cover pivots relative to the backplate and is pushed to pivot about a pivot point and actuate the pump. The pivot point is positioned such that the volume of the reservoir recess increases as the pump is actuated to dispense the product. The product reservoir may include a pair of cylindrical recess positioned on a common axis and projecting inwardly toward one another, and the cover may include a pair of inwardly projecting journals that are removably received in the cylindrical recesses, the cylindrical recesses and the journals acting together to form the pivot point of the cover.