We study the possibility that the vacuum energydensity of scalar and internal-space gauge fieldsarising from the process of dimensional reduction ofhigher dimensional gravity theories plays the role of quintessence. We show that, for themultidimensional Einstein-Yang-Mills system compactifiedon a R × S3 × Sdtopology, there are classically stable solutions suchthat the observed accelerated expansion of the Universe atpresent can be accounted for without upsetting structureformation scenarios or violating observational bounds onthe vacuum energy density.