Healthy older people with a cortisol awakening response (CAR) of decreased magnitude show worse frontal cortex-related cognitive performance. Systemic hypertension has been related to a CAR of decreased magnitude. Additionally, worse executive function and processing speed have been observed in older people with systemic hypertension. This is the first study to examine the relationship between the CAR (measured with six saliva samples at home on two consecutive weekdays) and cognitive performance, in both hypertensive (n = 26) and normotensive (n = 28) older people (from 56 to 78 years old). Hypertensive participants showed lower morning cortisol secretion, and they also woke up earlier. No differences in CAR were observed. A CAR of decreased magnitude was related to worse executive function in both hypertensive and normotensive participants, but to slower processing speed only in normotensive participants. Being treated with antihypertensive for a longer period of time was related to a CAR of increased magnitude and better performance on executive function. Our findings suggest that earlier awakening time in hypertensive older people might underlie the lower overall morning cortisol secretion observed in previous studies. Additionally, this study confirms that a dysregulation of the CAR is related to worse executive function, and it extends this association to hypertensive older people. Finally, it is worth noting that hypertension may moderate the relationship between CAR and processing speed.