The first song I can remember from hearing by The Pet Shop Boys was Willie Nelson's "Always On My Mind". The first RHCP song I remember hearing was Stevie Wonder's "Higher Ground". Soft Cell's "Tainted Love" is a cover of Gloria Jones' 1964 recording.
But I have a sense that at some point, this became a very deliberate strategy on the part of record companies promoting artists: familiar song, new artist's sound, change one thing only to make the new sound accessible.
https://www.youtube.com/watch?v=NSehtaY6k1U