Alzheimer’s disease (AD) is a multifactorial syndrome with a plethora of progressive, degenerative changes in the brain parenchyma, but also in the cerebrovascular and hemostatic system. A therapeutic approach for AD is reviewed, which is focused on the role of amyloid–β protein (Aβ) and fibrin in triggering intra-brain vascular dysfunction and connected, cognitive decline. It is proposed that direct oral anticoagulants (DOACs) counteract Aβ-induced pathological alterations in cerebral blood vessels early in AD, a condition, known as cerebral amyloid angiopathy (CAA). By inhibiting thrombin for fibrin formation, anticoagulants can prevent accumulations of proinflammatory thrombin and fibrin, and deposition of degradation-resistant, Aβ-containing fibrin clots. These fibrin–Aβ clots are found in brain parenchyma between neuron cells, and in and around cerebral blood vessels in areas of CAA, leading to decreased cerebral blood flow. Consequently, anticoagulant treatment could reduce hypoperfusion and restricted supply of brain tissue with oxygen and nutrients. Concomitantly, hypoperfusion-enhanced neurodegenerative processes, such as progressive Aβ accumulation via synthesis and reduced perivascular clearance, neuroinflammation, and synapse and neuron cell loss, could be mitigated. Given full cerebral perfusion and reduced Aβ- and fibrin-accumulating and inflammatory milieu, anticoagulants could be able to decrease vascular-driven progression in neurodegenerative and cognitive changes, present in AD, when treated early, therapeutically, or prophylactically.