Press won a gold medal in the 1960 Olympics in Rome in the 80-metre hurdles, setting an Olympic record (10.6 sec) in the semifinals. Her sister Tamara (Press, Tamara) also competed in Rome and won a gold medal in the shot put; in the process, they became the first sisters to win gold medals in the same Olympics. At the 1964 Olympics in Tokyo, Press finished fourth in the 80-metre hurdles, but she won her second gold medal in the pentathlon, which was a women's event for the first time in the 1964 Games. She had the best time or distance in two of the five events, tallying 5,246 points and setting a world record.

Throughout much of their careers, Irina and Tamara faced widespread speculation either that they were taking male hormones or that they were actually men. When the International Amateur Athletic Federation (later called the International Association of Athletics Federations) announced in 1966 that it would conduct sex testing of athletes competing in the upcoming European championships, both sisters withdrew from the competition. Neither competed in the Olympics after 1964.