For complex linguistic strings such as idioms, making a decision as to the correct meaning may require complex top-down cognitive control such as the suppression of incorrect alternative meanings. In the study presented here, we used transcranial direct current stimulation to test the hypothesis that a domain general dorsolateral prefrontal cognitive control network is involved in constraining the complex processing involved. Specifically, we sought to test prominent theoretical stances on the division of labour across dorsolateral prefrontal cortex in the left- and right-hemispheres of the brain, including the role of salience and fine vs. coarse semantic coding. 32 healthy young adult participants were randomly allocated to one of two stimulation montage groups (LH anodal/RH cathodal or RH anodal/LH cathodal). Participants were tested twice, completing a semantic decision task after either receiving active or sham stimulation. The semantic decision task required participants to judge the relatedness of an idiom and a target word. The target word was figuratively related, literally related, or unrelated to the idiom. Control non-literal non-idiomatic sentences were also included that only had a literal meaning. The results showed that left-hemisphere dorsolateral prefrontal cortex is highly involved in processing figurative language, whereas both left- and right- dorsolateral prefrontal cortex contributed to literal language processing. In comparison, semantic processing for the non-idiomatic control sentences did not require domain general cognitive control as it relates to suppression of the rejected alternative meaning. The results are discussed in terms of the interplay between need for domain general cognitive control in understanding the meaning of complex sentences, hemispheric differences in semantic processing, and salience detection.