Quirot was born in Palma Soriano, Cuba. In 1983, she won a silver medal in the 400 metres at the Pan American Games in Caracus, running 51.83. Four years later at the Pan American Games in Indianapolis, she won both the 400 m and 800 m. In the 400 m, she ran 50.27 to defeat Canada's Jillian Richardson, while in the 800 m she defeated Delisa Walton-Floyd of the USA in 1:59.06. Later that year at the 1987 World Championships in Rome, she improved her 800 m best 1:55.84, to finish fourth in a high quality final. The race was won by East Germany's Sigrun Wodars in 1:55.32.
At the 1989 IAAF World Cup in Barcelona, Quirot reached her peak at 800 metres. In a race that was fast from the start, thanks to the front running of World and Olympic champion Wodars, Quirot won in 1:54.44, to move to third on the world all-time list behind world record holder Jarmila Kratochvilova and 1980 Olympic champion Nadezhda Olizarenko. She also won the 400 m, after original winner Marie-Jose Perec was disqualified for running out of her lane. In 1990, she again achieved a 400m, 800 m double, this time at the Goodwill Games in Seattle. She won the 400 m in 50.38 and the 800 m in 1:57.42, narrowly ahead of the Soviet Union's Liliya Nurutdinova, who ran 1:57.52.
Quirot was unbeaten at the 800 metres for almost three years, from her fourth place at the 1987 Worlds, to the Zurich Grand Prix in August 1990, when she was third behind the East German pair of Wodars and Christine Wachtel. This lone defeat bumped her down the 3rd in the Track and Field News Magazine 1990 800 metre rankings, as it was her only meeting with the 2 women in 1990, after having been ranked 1st in both 1988 and 1989. Going in as favorite she settled for a silver medal at the 1991 1991 World Championships in Tokyo, narrowly losing to Nurutdinova 1:57.50 to 1:57.55. The first 4 finishers were within 0.13 of a second and closely crowded in each other's paths at the very end, with bronze medalist Kovacs sprawling across the line, blocking out the path of 4th-place finisher Maria Mutola who may have won the race otherwise. She did though manage a 4-1 record vs Nurutdinova for 1991, and a clear winning record vs all her main competitors (only 5 2nd place losses to 5 different women) and regained her #1 ranking over 800 metres in the final Track and Field News Magazine yearly rankings.
Having been prevented from competing at the 1984 and 1988 Olympic Games, due to the Cuban boycotts, Quirot made her Olympic debut at the 1992 games in Barcelona, where she won a bronze medal in the 800 m behind Ellen van Langen and Nurutdinova. Ana had gone into the race as the gold medal favorite, with Nurutdinova tipped as her biggest rival, and Ludmila Gurina, Ellen Kovacs, and young Maria Mutola also viewed as real contenders, but the overlooked Van Langen used a smart strategic race and devastating finish to steal victory from the favorites, slipping past a tiring Nurutdinova on the inside 50 metres from the finish.
In 1993, she was involved in a domestic accident that left her seriously injured. She was pregnant at the time and gave birth to her daughter prematurely in hospital while fighting for her life. Her daughter did not survive and died a week after she was born.
Quirot returned from her accident in late 1993 and won a silver medal in the Central America Games, behind the Surinam athlete Letitia Vriesde. Then in 1995, at the World Championships in Gothenburg, she became World champion for the first time ever, defeating Vriesde and Kelly Holmes, who were second and third respectively. In gaining victory she used her trademark blinding finishing kick to move from 5th to 1st down the stretch, something that comes largely from her background of also being a longtime top 400 metre runner. Maria Mutola was heavily favored, having been on a 3-year unbeaten streak, but stepped on a lanemarker in the semis, being disqualified. Mutola would be victorious in all head to head encounters between her and Ana over 800 metres this year, including the season ending grand prix final where Mutola would place 1st to Quirot's 5th.
Quirot won her second Olympic medal at the 1996 Olympic Games in Atlanta, winning silver. She went in as one of the top 2 favorites, along with Maria Mutola, and a battle between the 2 greats was widely anticipated, with Kelly Holmes also considered a possible gold medal contender. She ultimately defeated both Mutola and Holmes, but ended being pipped for the gold in a major upset by Svetlana Masterkova, who had returned to the sport in 1996, after two years away to have a baby. She was disappointed, knowing at age 33 she had likely missed her last chance of an Olympic Gold, but considered it a triumph after the adversity she had faced to get there. She blamed a poor tactical race on her defeat, and likely lost the race when she allowed Holmes to pass her on the backstretch, slipping back to 4th, rather than responding by moving onto the leader Masterkova's shoulder before the final bend. She would hand Masterkova her only defeat of the year at 800 metres a few weeks later. Track and Field news ranked her 2nd for the year at 800 metres, behind Masterkova, but ahead of Mutola.
In 1997, Quirot retained her World title at the 1997 World Championships in Athens, with Yelena Afanasyeva second and Mutola third. It was a far smarter tactical race for Quirot than her misjudged Atlanta race, staying close to the leader all the way and out of trouble, keeping a clear path for herself to the end, and utilizing her trademark finishing kick to lay waste to Mutola and the Russians. She also was named the No. 1 ranked 800 runner for the year for the 4th and final time, and her first time since 1991. She trailed her chief rival Mutola 4-3 in head to head meetings this year, but was ranked 1st due to winning the far more important encounters at the World Championship, Grand Prix Finale, and in going under 1:55 in Cologne.
Quirot is one of only six female athletes who have run under 1 minute and 55 seconds for 800 m. Her best time of 1:54.44 from 1989, ranks her fourth on the world all-time list behind Kratochvilova, Olizarenko and Pamela Jelimo. She also ran 1:54.82 to beat Maria Mutola in a Grand Prix race in Cologne in 1997. Her lifetime best for 400 m is 49.61 in 1991. She also holds the world best for the unofficial distance of 600 metres with 1:22.63 at altitude in 1997. In her final year of 1997 she would be ranked No. 1 in the world over 800 metres by Track and Field News Magazine, a feat she had last managed way back in the years 1988, 1989, and 1991. Mutola won 4 of 7 encounters with Quirot that year, but Quirot with the more prestigious victories- World Championships, her sub 1:55 clocking, and the IAAF Grand Prix final, was rightly given the nod by the voters. Upon her return to running after her near deadly accident, she had been ranked #2 in both 1995 (behind Mutola) and 1996 (behind Masterkova).
One of her most impressive career stats is a perfect 4-0 record against the great Maria Mutola, widely considered the greatest 800 metre runner in history, in World and Olympic competition. She cracked the 1:55 barrier on 2 occasions, and was the only female to break 1:55 between over a 25-year span from 1983 (Kratochvílová) to 2008 (Jelimo). She is one of a handful of women to ever be ranked No. 1 in the world in both 400 and 800 metres in the same year, a feat she achieved in her career year of 1989, when she was named IAAF track athlete of the year by the prestigious Track and Field News Magazine. She was quite possibly denied her elusive Olympic Gold in the 800 metres in 1988, and a medal in the 400 metres, had it not been for the boycott, as she was unbeaten in the 800 metres all year, and a world leader in the 400 metres.
Along with Jarmila Kratochvilova (who is still today the world record holder at 800 metres), she is wildely regarded as the greatest 800 metre runner without an Olympic Gold medal.