The cascade of electronic phases in CsV3Sb5 raises the prospect to disentangle their mutual interactions in a clean, strongly interacting kagome lattice. When the kagome planes are stacked into a crystal, its electronic dimensionality encodes how much of the kagome physics and its topological aspects survive. The layered structure of CsV3Sb5 reflects in Brillouin-zone-sized quasi-two-dimensional Fermi surfaces and significant transport anisotropy. Yet here we demonstrate that CsV3Sb5 is a three-dimensional (3D) metal within the charge density wave (CDW) state. Small 3D pockets play a crucial role in its low-temperature magneto-and quantum transport. Their emergence at T-CDW asymptotic to 93 K results in an anomalous sudden increase of the in-plane magnetoresistance by four orders of magnitude. The presence of these 3D pockets is further confirmed by quantum oscillations under in-plane magnetic fields, demonstrating their closed nature. These results emphasize the impact of interlayer coupling on the kagome physics in 3D materials.