The Nobel-winning scientist, then 42, wrote to Elisabetta Piccini, a 22-year-old chemistry student.
Ms Piccini lived one floor above his sister, Maja, in Florence.
"Einstein was very interested in meeting her. [But] Ms Piccini was too shy to meet with such a famous person" Winner's auction house said.
In the letter written in his native German, Einstein uses an idiom signalling affection.
"To the scientific researcher, at whose feet I slept and sat for two full days, as a friendly souvenir," the note to Ms Piccini said.
"You know nowadays the 'Me Too' campaign? Probably Einstein would have been in this campaign by leaving such a note to this lady," Gal Wiener, chief executive of Winner's, told the Associated Press news agency.
The note was sold alongside a number of other letters from Einstein, including a 1928 note in which Einstein set out his thoughts for his "Third Stage of the Theory of Relativity" which was auctioned off for $103,000 (£74,000).
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