Stephens saved five match points serving at 5-6, sending it to a tiebreaker where Kirilenko clinched the match after a wide forehand by Stephens.
An American seeded 18th, Stephens had held the longest active run of fourth-round appearances at majors by a woman, dating to a semifinal showing at the 2013 Australian Open.
At Wimbledon last year, Stephens got to the quarterfinals, where she lost to eventual champion Marion Bartoli.
Coming into Monday, Stephens’ career record in Grand Slam matches was 31-12, while she was 55-54 everywhere else.