(1) Background: The aim was to measure the efficiency and productivity of 15 specialty clinics during the COVID-19 pandemic period 2020–2021 in the General Hospital of Rhodes. (2) Methods: An input-oriented data envelopment analysis and the Malmquist productivity index are used. Labor and capital were used as inputs, and in-patient discharges and days were used as outputs. (3) Results: Five out of the seven clinics in the pathology sector appeared fully efficient with an optimal productivity, and the rest showed progress in 2021. In 2020 the COVID-19 pathology clinic appeared to be inefficient and less productive, while in 2021, it showed a positive performance change. The surgical sector showed very high efficiency rates or even reached an optimal efficiency in both years. The productivity measurement, in most of the surgical clinics, was satisfactory to very high. In 2020 the COVID-19 surgical clinic appeared to be more efficient and productive than in 2021 when its performance declined. (4) Conclusions: The hospital responded to the pressure during the pandemic, by increasing its efficiency and productivity from 2020 to 2021. This was due to the accomplishment of the appropriate organizational changes in the infrastructure, human resources, and technology. The efficiency and productivity assessments should be incorporated in the hospitals’ decision making.