"Godson of Almarlu", a novella by Raymond Z. Gallun in Astounding Stories, October 1936, available at the Internet Archive. You probably read it in the 1978 collection The Best of Raymond Z. Gallun.
From Everett F. Bleiler's review in Science-Fiction: The Gernsback Years:
In a short prologue a mechanical device much like that described in #471, "The Scarab" comes to Earth, makes its way into a house, and injects a substance into a sleeping baby. This is not explained at the time, but it is part of a long-range plan dating from before the rise of humanity.
Background, as imparted gradually: There is a rogue planet, apparently composed mostly of neutronium, that makes a periodic swing through the solar system on an orbit of perhaps hundreds of millions of years. On a previous pass it destroyed Bode's fifth planet (Almarlu), which then held a supercivilization. The natives of Almarlu did not have time to save their own world, but they distributed seeds of their biota, including themselves, on the as yet lifeless Earth. They also set up a monitor to prepare for the next approach of the neutronium planet. Such a monitor selected baby Jeff Scanlon and prepared him for the task of saving mankind.
Jeff, who seems otherwise an average man, demonstrates incredible financial wizardry, and in a short time is financially the most powerful man on Earth. In semiretirement, he betakes himself to science, and, obviously drawing unconsciously on the knowledge of Almarlu, about which he occasionally dreams, creates communications marvels and wireless transmission of power.
His chef d'oeuvre, however, is a device for tapping the Earth's rotational inertia. Jeff cannot explain what he has done, nor does he understand it, but it works. He builds an incredibly strong power station on a small subarctic island. This station is so massively built that it is practically indestructible.
At just this time, the neutronium planet becomes visible. Almarlu has predicted well. But when Jeff turns on his apparatus full force, focused on the Moon, it drives air and water out into space. Earthquakes, tidal waves, submersions begin to take place.
Jeff is blamed for the damage, and his position is not enviable. But this is only a preparatory step. Jeff broadcasts that the Earth is doomed. He urges everyone to take to the air and fly into the beam radiating from his station. The beam will carry them to the Moon and safety. He is not generally believed, but enough people follow his advice that the Moon has a substantial population, transported along with air and water from Earth by Jeff's beam. As the survivors watch, Earth splits apart and dies.
In Astounding the story was illustrated with three drawings by C. R. Thomson.