Single nutrients or food groups have been associated with physical performance. However, little is known about the association of overall diet quality with hand grip strength (HGS), a predictive parameter in the prognosis of chronic disease morbidity and mortality, or quality of life. This study examined the association between HGS and three indices—the Korean Healthy Eating Index (KHEI), the Alternate Mediterranean Diet (aMED), and Dietary Approaches to Stop Hypertension (DASH)—using data obtained on Korean elderly persons aged ≥65 years (n = 3675) from a nationally representative database. The cross-sectional data was collected as part of the Korea National Health and Nutrition Examination Survey (KNHANES, 2014–2016). Dietary intake data from the 24-h recall method were used to calculate diet quality scores. The cutoff value for low HGS was defined as the value corresponding to the lowest 20th percentile of HGS of the study population (men, 28.6 kg; women, 16.5 kg). Higher index scores for diet quality were associated with 32%–53% lower odds of low HGS. Better overall diet quality may be associated with higher HGS in the elderly Korean population.