In the first episode of the MTM show, Rhoda claimed Mary's apt
was hers. While in the end we know Mary got to keep it, how
did Mary, a newcomer, prevail over Rhoda who was already living
in the building?

because Phyllis rented it out to her friend Mary.

because Phyllis rented it out to her friend Mary.


That's it! However; it should be noted that Phyllis kept talking about 'the landlady' as though it was someone other than her. And it wouldn't be until a later episode that Phyllis admitted that 'L&P Management' stood for herself and Lars.

Of course, it's interesting that even though Mary and Rhoda had barely truced it over by the end of the first episode, by the 2nd episode they were TOTALLY best pals to the point of trying to up the odds nabbing husbands by having a double-date in Mary's apartment (to say it backfired is an understatement but what would have been the comedy had they both met Prince Charmings and lived happily ever after only at the 2nd episode).

Ironically enough, a decade later on "Cheers", they started out Diane Chambers and Carla on the SAME sour note opposites-barely trucing it over note but, unlike Mary and Rhoda, these two NEVER became more than grudging trucemates.