Kurzak was born to a musical family in Brzeg Dolny, Poland, on August 7, 1977; her father is a French horn player, and her mother is soprano Jolanta Zmurko. Aleksandra considers her mother her greatest teacher and credits Zmurko with developing her vocal technique, despite having many other teachers in her student years. At seven, Kurzak began music studies on both the violin and piano, but her main focus would turn to the violin. She also harbored hopes for a career as a ballet dancer. It was not until she was 19 that Kurzak began vocal studies. From 1996-2000, she studied at the Wroclaw Conservatory in Poland. During this time, she was also busy in competitions and on the operatic stage: in 1998, Kurzak won the Stanislaw Moniuszko International Vocal Competition, and the following year, she debuted at the Wroclaw State Opera as Susanna in Mozart's Le nozze di Figaro. Kurzak also won other competitions, including Barcelona's Viñas International Singing Competition in 2000.
From 2001-2007, she sang with the Hamburg State Opera, first as an apprentice in the company's Young Artist program, then, from 2003, as a member. While she initially sang less important roles there, such as Kate Pinkerton in Puccini's Madama Butterfly, she eventually took on meatier fare, including Gilda in Verdi's Rigoletto and Mozart's Queen of the Night from The Magic Flute. During this time, Kurzak also studied at the Hochschule für Musik in Hamburg. Meanwhile, invitations to the major opera houses came one after the other. Kurzak debuted at the Met in 2004 as Olympia in Offenbach's Les contes d'Hoffmann. In 2005, she debuted at Covent Garden as Aspasia in Mozart's Mitridate, rè di Ponto. Kurzak had several return engagements at the Met, including as Blonde in Mozart's Abduction from the Seraglio (2008), and as Gretel in Humperdinck's Hansel and Gretel (2011). Her La Scala debut in 2010 was also as Gilda.
Kurzak signed an exclusive recording contract with the Decca label in 2010, and her first recording, Gioia, containing arias by Mozart, Rossini, Verdi, Puccini, and others, was issued in 2011. She has also been featured on releases from Warner Classics, Deutsche Grammophon, and Sony Classical. In 2017, Kurzak married tenor Roberto Alagna and was featured on Alagna's 2019 album Caruso 1873. In 2020, Kurzak released an album of opera arias on Sony Classical titled Desire. ~ Robert Cummings, Rovi