In a device for cleaning an inkjet print head (in particular the surface with nozzle exit openings) for a franking and/or addressing machine in which the print substrates are directed along a stationary but pivotably arranged inkjet print head, the entire nozzle exit surface is cleaned in a short time and a contamination of the transport region is avoided, with a small apparatus expenditure, by a wiping device formed by a driven wiping roller that is transversally, non-positively directed along the nozzle exit surface in a cleaning operation, and that continuously, non-positively rests on an associated cleaning element. The nozzle surface is repeatedly wiped off and the wiping roller is simultaneously cleaned in a single pass. An actuator of the wiping roller can be selectively coupled with an actuator for a cleaning and sealing device that operates the cleaning element, or can be autonomous. The cleaning element can be a stripper for the wiping roller that rests parallel to and positively on the wiping roller.