Transcranial direct current stimulation (tDCS) targeting the prefrontal cortex reduces the size and speed of standing postural sway in younger adults, particularly when performing a cognitive dual task. Here, we hypothesized that tDCS would alter the complex dynamics of postural sway as quantified by multiscale entropy (MSE). Twenty healthy older adults completed two study visits. Center-of-pressure (COP) fluctuations were recorded during single-task (i.e., quiet standing) and dual-task (i.e., standing while performing serial subtractions) conditions, both before and after a 20-min session of real or sham tDCS. MSE was used to estimate COP complexity within each condition. The percentage change in complexity from single- to dual-task conditions (i.e., dual-task cost) was also calculated. Before tDCS, COP complexity was lower (p = 0.04) in the dual-task condition as compared to the single-task condition. Neither real nor sham tDCS altered complexity in the single-task condition. As compared to sham tDCS, real tDCS increased complexity in the dual-task condition (p = 0.02) and induced a trend toward improved serial subtraction performance (p = 0.09). Moreover, those subjects with lower dual-task COP complexity at baseline exhibited greater percentage increases in complexity following real tDCS (R = -0.39, p = 0.05). Real tDCS also reduced the dual-task cost to complexity (p = 0.02), while sham stimulation had no effect. A single session of tDCS targeting the prefrontal cortex increased standing postural sway complexity with concurrent non-postural cognitive task. This form of noninvasive brain stimulation may be a safe strategy to acutely improve postural control by enhancing the system's capacity to adapt to stressors.