If you are not looking for philosophical reasons that Marxism dominated socialist thought in the twentieth, but simply the historical path that the ideology followed, then the answer is actually really simple: Lenin and the Bolsheviks, who were Marxists, seized control of socialist discourse after they seized control of Russia in the October Revolution. Prior to that, Marxism had not been the dominant ideology among socialists, but rather the more moderate "social democracy" exemplified by the German SPD. Marxism became the ideology of post-revolutionary Russia (Lenin made sure of this) and subsequently the Soviet Union, which then influenced Communists throughout Europe as the only socialist power in existence. This influence only grew with the creation of the Soviet sphere and the rise of Soviet prestige in the wake of the Second World War.

If you want to know why revolutionaries like Lenin (but not only Lenin and not only Russians/Soviets) embraced Marxism from a philosophical standpoint, my answer will be far more subjective, though I think it's very sound to argue that the revolutionary language of Marx appealed to people who wanted to enact revolutions. Lenin despised traditional social democracy exactly because, as he noted his 1901 treatise "What Is To Be Done," it trapped workers in a "trade-unionist" mindset that ultimately served the bourgeoisie. Instead, it should be the task of revolutionaries like himself to "bring [the workers] under the wing of revolutionary Social Democracy" [note that the terms "Bolshevik" and "Communist Party" did not yet exist at this time]. The rhetoric of Marx, which called for revolution, satisfied the desire of revolutionaries. Moreover, Marx's assertion that his theory was not ideological but "scientific" appealed to Lenin. Marx had discovered a natural system of progress, and it just so happened to call for revolution. This was the perfect mix for the self-identified intellectual revolutionary.

We also see the influence of Marxist rhetoric clearly in the case of postwar Czechoslovakia, where many young Communists once again embraced Marx as a guide in overthrowing the bourgeois society that had allowed the Germans to cut a deal with the British to dissect Czechoslovakia in 1938 (Czechoslovakia being one of the rare examples where a domestic Communist movement played the decisive role in that nation's slide into the Soviet sphere). I think this point is important because it provides a completely separate example of how Marxist rhetoric appealed to those who suffered through catastrophe and who desired fundamental--and if necessary, violent--changes in the structure of society.