As a junior, Zvereva won the Wimbledon girls’ singles title in 1986, defeating Leila Meskhi in the final 2–6, 6–2, 9–7. Zvereva also won the US Open girls singles championship in 1987, beating Sandra Birch in the final 6–0, 6–3.

Although Zvereva was a highly accomplished doubles player, and considered by some to be one of the best doubles players of all time, she had limited success in singles. Nevertheless, in 1988, at the age of 17, she made her sole Grand Slam singles final at the French Open beating Martina Navratilova en route. In a highly publicized final, she lost to Steffi Graf 0–6, 0–6 in only 34 minutes. The official time of the match given on the scoresheet was 34 minutes, however just 32 minutes of that was spent on the court, as a rain break split the match into two periods of play, of nine and 23 minutes. This was the shortest and most one-sided Grand Slam final ever. (Graf went on to win all four Grand Slam singles titles and an Olympic gold medal that year.)

In addition to her Grand Slam women’s doubles titles, Zvereva twice won the mixed-doubles title at the Australian Open. She partnered with Jim Pugh to win the title in 1990, and with Rick Leach in 1995.

Meskhi and Zvereva lost in the semifinals to Gigi Fernández and Mary Joe Fernández 4–6, 5–7. In 1992, there was no bronze medal play-off match, both beaten semifinal pairs received bronze medals.

Zvereva was born as Natalya Marataŭna Zvereva in Minsk, Belarus, to parents Marat Zverev and Nina Zvereva. She started tennis at the age of seven at the encouragement of her parents, who were both tennis instructors in the Soviet Union. While her name is sometimes spelled Zverava, in 1994 she officially changed her name to Natasha Zvereva.

Zvereva is one of the few players to have beaten both Graf and Monica Seles, both former world number ones, in the same Grand Slam singles tournament. Ten years later at Wimbledon in 1998, Zvereva finally defeated the fourth seeded Graf in the third round 6–4, 7–5 and also the sixth seeded Seles in a quarterfinal 7–6, 6–2. This was also notable because it was Zvereva’s sole win over Graf in 21 career singles matches. She lost in the semifinals to Nathalie Tauziat which was to be her second best career Grand Slam singles result. Starting with the French Open in 1987 and extending through Wimbledon in 2000, Zvereva played in 51 of the 54 Grand Slam singles tournaments held, reaching the quarterfinals or better eight times.

Zvereva retired from professional tennis in 2003. Her last appearance in a Grand Slam tournament was in Wimbledon 2002, where she lost in the first round to Marlene Weingärtner 6–4, 3–6, 2–6. She played in the invitational doubles event in 2007.