Normotensive and hypertensive organisms respond differently to stress factors; however, the features of the central molecular genetic mechanisms underlying the reaction of the hypertensive organism to stress have not been fully established. In this study, we examined the transcriptome profiles of the hypothalamus of hypertensive ISIAH rats, modeling a stress-sensitive form of arterial hypertension, and normotensive WAG rats at rest and after exposure to a single short-term restraint stress. It was shown that oxidative phosphorylation is the most significantly enriched process among metabolic changes in the hypothalamus of rats of both strains when exposed to a single short-term restraint stress. The analysis revealed DEGs representing both a common response to oxidative stress for both rat strains and a strain-specific response to oxidative stress for hypertensive ISIAH rats. Among the genes of the common response to oxidative stress, the most significant changes in the transcription level were observed in Nos1, Ppargc1a, Abcc1, Srxn1, Cryab, Hspb1, and Fosl1, among which Abcc1 and Nos1 are associated with hypertension, and Fosl1 and Ppargc1a encode transcription factors. The response to oxidative stress specific to hypertensive rats is associated with the activation of the Fos gene. The DEG’s promoter region enrichment analysis allowed us to hypothesize that the response to oxidative stress may be mediated by the participation of the transcription factor CREB1 (Cyclic AMP-responsive element-binding protein 1) and the glucocorticoid receptor (NR3C1) under restraint stress in the hypothalamus of both rat strains. The results of the study revealed common and strain-specific features in the molecular mechanisms associated with oxidative phosphorylation and oxidative stress response in the hypothalamus of hypertensive ISIAH and normotensive WAG rats following a single short-term restraint stress. The obtained results expand the understanding of the most significant molecular targets for further research aimed at developing new therapeutic strategies for the prevention of the consequences of acute emotional stress, taking into account the hypertensive state of the patient.