The American 11th seed, who triumphed here in 2017, fought off two match points but on the third was caught out of position behind the baseline when the Russian ripped a backhand that clipped the net chord and landed in.
"I was playing one good point, one bad point, one good point. The inconsistency doesn't help me at all," said Stephens.
- Kyrgios keeps mostly calm and carries on at U.S. Open
- Muguruza crashes out of US Open, Osaka, Kvitova and Halep through
"She did a good job just staying there, keep fighting. She never really let me in. Kudos to her. She executed her game style really well."
Kalinskaya, playing her first career Grand Slam night match and first inside Arthur Ashe Stadium, did not appear intimidated facing Stephens, whose U.S. Open tune-up included a run to the last 16 in Cincinnati.
The 20-year-old Russian, who won three qualifying matches to make the main draw, broke Stephens at her first opportunity to pull ahead 5-3 in the first set and never looked back as she went on to secure the biggest win of her young career.
"It feels amazing to win the first match on this court, a subdued Kalinskaya said in her on-court interview. "I just tried to focus and don't think too much about her and think just about my game."
Kalinskaya was only broken once during the match and managed to convert three of her four break point chances.
Up next for Kalinskaya will be American wildcard Kristie Ahn, who stunned Cincinnati runner-up and 2004 U.S. Open champion Svetlana Kuznetsova earlier on Tuesday.