Egerszegi was only 14 years old when she competed at the 1988 Olympic Games in Seoul. Emboldened by the silver medal she gained in the 100m backstroke, she felt that she could upset the favoured East Germans at 200m, even though, at 45kg, she was 19kg lighter than any of her opponents in the final. Egerszegi sprinted into the lead after the midrace turn and won going away.
In 1992, Egerszegi, by now a seasoned 18-year-old, won both backstroke events. She also entered and won the 400m individual medley, recording the fastest time since Petra Schneider's chemically augmented 1982 world record.
At the 1996 Olympic Games in Atlanta, Egerszegi earned a bronze medal in the 400 IM and then won the 200m backstroke to become only the second swimmer in history to win the same event three times. Her margin of victory, 4.15 seconds, was the greatest in any women's 200m event. In 1996, Egerszegi did not enter the 100m backstroke, however her leadoff backstroke time in the medley relay, 1:01.15, was faster than the winning time in the 100m backstroke final.