The Lithuanian pair finished the distance in 32.32 seconds.
Lankas, 30, and Ramanauskas, who celebrates his 31st birthday on Thursday, were leading the race for most of the distance, but were overtaken by Spanish and British athletes at the very end.
The Olympic gold went to Saul Craviotto Rivero and Cristian Toro Carballo (32.075 seconds) of Spain, while Britain's John Shofield and Liam Heath secured silver (32.368 seconds).
Hungarian, German, Serbian, French and Canadian athletes were left outside the top three.
This is Lithuania's fourth Olympic medal in Rio Games. Rowers Mindaugas Griškonis and Saulius Ritter won silver medal, while the pair of Donata Vištartaitė and Milda Valčiukaitė were third in women's rowing event. Weightlifter Aurimas Didžbalis has also won a bronze medal.