Hellhounds of the Cosmos

Story, first published: Astounding Stories, June 1932

It's the story of an inter-dimensional invasion by an alien species - but the "hellhounds" turn out not to be extraterrestrials - but rather, earth creatures from an unimaginably distant past -- and another plain of existence!

Ken Korczak: Amazon Customer Review

The Horrors strike in many places around the world; black, amorphous shapes, they are completely invulnerable to all weapons-explosives, gas, radiation. Their numbers increase, and it looks as if mankind is doomed. Eccentric, even slightly mad Professor Silas White, believing that he has a solution to the problem, asks that a reporter be assigned to interview him. White then explains the situation to young reporter Henry Woods. The Horrors are three-dimensional versions of four-dimensional beings from another frame of existence, whence their invulnerability. White adds that he has created a device to fight them: something that will change a three-dimensional person into a four-dimensional one. And he has assembleda band of ninety-nine volunteers who will undergo the change, be transformed into four-dimensional beings in the four-dimensional world, and combat the monsters. Henry agrees to accompany the party.
When Henry finds himself in the strange, four-dimensional world, however, he discovers that he is part of a multiple being composed of all the ninety-nine other men, and that he has a former fourth-dimensional identity (Mal Shaf) that he is resuming. And the Horrors, similarly, are three-dimensional projections of another fourth-dimensional being, Ouglat, who ages before had forced Mal Shaf out of his world. The two beings fight desperately, with changing fortunes, as Ouglat withdraws his projections from our world to gain strength. Eventually Mal Shaf wins, but does not return to our world. He remains with his fourth-dimensional fellows.

Bleiler, Everett F.: Science-Fiction - The Gernsback Years, p.382