Nursing burnout is associated with reduced nursing performance outcomes. Positive psychological capital is known to play an important role in improving workers’ job performance. However, the association among the three variables has rarely been addressed. The purpose of this cross-sectional descriptive study was to explore the association between burnout and nursing performance outcomes among Korean nurses working at a tertiary hospital and the mediating role of psychological capital in this relationship. Recruited through convenience sampling, a total of 285 nurses provided data on their demographic characteristics and completed a structured questionnaire consisting of items from the Professional Quality of Life Scale (burnout), Nursing Performance Scale, and Psychology Capital Questionnaire. Descriptive statistics, student’s t-tests, one-way analysis of variance, Pearson’s correlation coefficients, and multiple linear regression analyses were used to analyze data. The significance of the mediation effect was obtained using a bootstrap approach with the PROCESS macro. The mean age of participants was 30.51 years, and most participants were females (94.0%) and unmarried (71.6%); more than half (57.5%) experienced a severe workload. The average (±standard deviation) scores of burnout, nursing performance outcomes, and positive psychological capital were 28.77 ± 4.93, 2.98 ± 0.32, and 3.19 ± 0.45, respectively. Burnout was associated with nursing performance among clinical nurses (β = −0.20, p < 0.001). Positive psychological capital mediated the association between burnout and nursing performance outcomes (β = 0.41, p < 0.001). These findings contribute to the understanding that burnout among nurses could be reduced by increased positive psychological capital, which results in improved performance outcomes. The findings also indicate that interventions to improve positive psychological capital should be developed and implemented for nurses’ burnout management and improvement in nursing performance outcomes.