The solvation and the onset of dissolution of a cellulose I beta microcrystal in ionic liquid media are studied by molecular simulation. Ionic liquids can dissolve large amounts of cellulose, which can later be regenerated from solution, but their high viscosity is an inconvenience. Hydrogen bonding between the anion of the ionic liquid and cellulose is the main aspect determining dissolution. Here we try to elucidate the role of a molecular cosolvent, dimethyl sulfoxide (DMSO), which is an aprotic polar compound, in the system composed of cellulose and the ionic liquid 1-butyl-3-methylimidazolium acetate. We calculated quantities related to specific interactions (mainly hydrogen bonds), conformations, and the structure of local solvation environments, both for a solvated oligomer chain of cellulose and for a model microfibril composed of 36 chains in the 1b crystal structure. We compare two solvent systems: the pure ionic liquid and a mixed solvent with an equimolar composition in ionic liquid and DMSO. All entities are represented by detailed all-atom, fully flexible force fields. The main conclusions are that DMSO behaves as an "innocent" cosolvent, lowering the viscosity and accelerating mass transport in the system, but without interacting specifically with cellulose or disrupting the interactions between cellulose with the anions of the ionic liquid. An understanding of solvation in mixed solvents composed of ionic liquids and molecular compounds can enable the design of high-performance media for the use of biomass materials.