Moller's first international competition was the 1974 British Commonwealth Games at Christchurch, where she finished fifth in the 800 m. Her time of 2:03.63 was her lifetime best and is still the fastest ever by a New Zealand junior (under 20) woman.

Although Moller ran her first marathon in 1979, there were no sanctioned marathons for females at an international athletics competition until 1984. She ran her first marathon on 23 June 1979, winning Grandma's Marathon in Duluth, Minnesota in 2:37:37. She then won her next 7 marathons. She was a triple winner of the Osaka Ladies Marathon, and in 1984 won the Boston Marathon.

All of Moller's four appearances at the Olympic Games were in the marathon. She won the Bronze medal in 1992 running 2:33:59

In the 1993 New Year Honours, Moller was made a Member of the Order of the British Empire, for services to athletics.In 2012 she was inducted into the Boulder (Colorado) Sports Hall of Fame. She has worked with the Lydiard Foundation and the Master Plan training system to share the lessons of running coach Arthur Lydiard.