Nurses’ work motivation impacts their job satisfaction and work engagement, affecting their quality of care. Work motivation, a personal resource, can be categorized into intrinsic and extrinsic motivation, each of which may function differently in the job demands–resources (JD–R) model. To study the effect of nurses’ intrinsic and extrinsic work motivation on work engagement in long-term care (LTC) facilities, we randomly selected 1200 facilities from 6055 LTC facilities in eastern Japan. Two nurses from each facility completed a self-report questionnaire—newly developed for this study for evaluating intrinsic and extrinsic work motivation—to assess their work engagement, job satisfaction, and work motivation. Multiple regression analysis of 561 valid questionnaires investigated the relationship between work motivation and work engagement, indicating that intrinsic work motivation, job satisfaction, and age had a significant positive effect on work engagement, while extrinsic work motivation had no significant effect. However, half the nurses chose to work because of extrinsic work motivation, explaining the high turnover rate of nursing staff in LTC facilities. Findings indicate the importance of measures to foster nurses’ intrinsic motivation to improve work engagement. Further research should investigate how to improve the intrinsic motivation of nurses working in LTC facilities.