The considerable marketing muscle behind “Super Mario Run” propelled Nintendo Co.’s first smartphone action game to the top of Apple Inc.’s app-store charts.

Industry trackers put first-day downloads for “Super Mario Run” in the millions, rivaling the popularity of “Pokémon Go,” which had been the hottest launch of the year. SuperData Research Inc. said the Mario game amassed nearly 5 million world-wide, at the higher end of estimates.

Nintendo...