Egerszegi made her international debut at the 1987 European Aquatics Championships at the age of 13, coming fourth in the 200 m backstroke and fifth in the 100 m backstroke. At the 1988 Summer Olympics she won silver medal in the 100 m backstroke and became Olympic champion in the 200 m backstroke. At the age of 14 years and 41 days she became the youngest-ever female Olympic champion in swimming. This youth record was broken in 1992 by Kyoko Iwasaki of Japan, who won a gold medal in the 200 meter breaststroke at the 1992 Barcelona Olympic Games at the age of 14 years and six days. At the end of 1988, she was the top-ranked swimmer in 200 m backstroke in the world ranking as well as she became the Swimmer of the Year and the Best Female Athlete of the Year in Hungary.