Rita Moreno wins Best Supporting Actress Academy Award 1961

At what would be the beginning of a stellar career with multiple awards, Rita Moreno won the Academy Award for Best Supporting Actress for her performance in West Side Story at the 34th annual ceremony held April 9, 1962. 

West Side Story was the big winner of the evening, winning ten awards out of eleven nominations.  Besides Moreno's win, it won:

Best Motion Picture

Best Directing for Robert Wise and Jerome Robbins

Best Supporting Actor for George Chakiris

Best Music (Scoring of a Musical Picture) for Saul Chaplin, Johnny Green, Sid Ramin, and Irwin Kostal

Best Sound for Gordon Sawyer and Fred Hynes

Best Art Direction (Color) for Boris Leven and Victor Gangelin

Best Cinematography (Color) for Daniel Fapp

Best Costume Design (Color) for Irene Sharaff, beating Edith Head.

Best Film Editing for Thomas Stanford

West Side Story's only loss was for Best Writing (Screenplay-Based on Material from Another Medium).

Breakfast at Tiffany's was nominated for five awards but won only two, both going to Henry Mancini: Best Music (Music Score of a Dramatic or Comedy Picture) and Best Music (Song) for Moon River with Johnny Mercer who wrote the lyrics. 

Henry Mancini wins 5 Grammy Awards 1962

Henry Mancini won five Grammy Awards for his work on Breakfast at Tiffany's at the 4th annual ceremony, held May 29, 1962.  His composition of the soundtrack won Best Sound Track Album or Recording Score from Motion Picture or Television, and Best Performance by an Orchestra - for Other Than Dancing.  The latter award was from the Pop category?

His original song Moon River won Record of the year, Song of the Year, and Best Arrangement.

West Side Story won Best Sound Track Album or Recording of Original Cast from a Motion Picture or Television.

Ray Charles won Best Rhythm & Blues Performance for Hit the Road Jack.