Certain new analytic solutions for slowly-rotating charged perfect-fluid universes coupled with zero-mass scalar field are found out to substantiate the possibility of the existence of rotating cosmological objects of such nature and their dynamics is investigated. The nature and role of the metric rotation Ω(r, t) as well as that of the matter rotation ω(r, t) under different conditions are studied. The effects of the charged field and the scalar field on the rotational motion are also discussed. In some solutions we find out the temporal restrictions on the models for real astrophysical situations. Rotating models which are expanding as well are obtained, in which cases the rotational velocities are found to decay with the time, and these models may be taken as good examples of real astrophysical objects in this Universe.