Background/Objectives: Occupational burnout poses a significant burden to healthcare personnel, institutions, and service users. Anesthesia technologists and technicians work in the shadow of the perioperative team, and a lack of attention to anesthesia support personnel may pose a significant risk to their wellbeing and the quality of care delivered. To date, only a few studies have investigated the prevalence of burnout among anesthesia technologists and technicians worldwide and in Saudi Arabia. Thus, the aim of this cross-sectional study was to assess the prevalence and contributing factors to burnout among anesthesia technologists and technicians in a single tertiary hospital in Saudi Arabia. Methods: The study utilized the Maslach Burnout Inventory-Human Services Survey (MBI-HSS) to assess burnout and a 10-point scale to assess stress levels. The MBI-HSS inventory consists of three subscales: emotional exhaustion (EE), depersonalization (DP), and personal accomplishment (PA). Univariate and multivariate linear regression analyses were used to identify correlates of each burnout subscale. Gender was included in the multivariable regression analysis in addition to significant variables from univariable analysis. Results: A total of 89 participants completed the survey. Based on each subscale of the MBI-HSS, more than 60% of the participants reported high to moderate EE, and more than half reported high to moderate DP. For PA, only 25.8% of participants reported low PA. We found that age (β = −0.58, 95% CI: −0.95, −0.20; p = 0.003) and stress (β = 3.3, 95% CI: 2.1, 4.5; p < 0.001) were independently associated with EE. In addition, night shift (β = 3.3, 95% CI: 0.44, 6.1; p = 0.024) and stress (β = 0.73, 95% CI: 0.13, 1.3; p = 0.017) were independently associated with DP. Independent factors for PA were identified including night shifts (β = 6.6, 95% CI: 1.4, 12; p = 0.014) and stress levels (β = −1.3, 95% CI: −2.4, −0.12; p = 0.03). Conclusions: This research underscores the alarmingly high prevalence of burnout and the strong link between elevated EE and DP rates and workplace stress, emphasizing the necessity to identify and mitigate these stressors. It is therefore crucial to evaluate the effectiveness of the current wellbeing and mental health initiatives and programs in Saudi Arabia to ensure that they address evolving challenges and the overall mental health of healthcare personnel.