Lithium niobate (LN) has been the material of choice for electro-optic modulators owing to its excellent physical properties. While conventional LN electro-optic modulators continue to be the workhorse of the modern optoelectronics, they are becoming progressively too bulky, expensive, and power-hungry to fully serve the needs of this industry. Here, we demonstrate plasmonic electro-optic directional coupler switches consisting of two closely spaced nm-thin gold nanostripes on LN substrates that guide both coupled electromagnetic modes and electrical signals that control their coupling, thereby enabling ultra-compact switching and modulation functionalities. Extreme confinement and good spatial overlap of both slow-plasmon modes and electrostatic fields created by the nanostripes allow us to achieve a 90% modulation depth with 20-μm-long switches characterized by a broadband electro-optic modulation efficiency of 0.3 V cm. Our monolithic LN plasmonic platform enables a wide range of cost-effective optical communication applications that demand μm-scale footprints, ultrafast operation and high environmental stability.