Dodt was born in Queensland, Australia and learned to play golf from the age of four at Gatton Golf Club, close to where he lived. He was a member of the Golf Australia National Squad and won many high profile amateur tournaments including the 2007 Australian Amateur Stroke Play. That win gave Dodt a spot in the 2007 MFS Australian Open at The Australian Golf Club in December. He turned professional at the conclusion of that tournament, and joined the PGA Tour of Australasia. He joined the Asian Tour a year later for the 2008 season. Since turning professional, his best result in Australia was second place at the 2010 New Zealand Open. Since then, Andrew won his second European Tour event at the True Thailand Classic. Due to good form and a top 3 finish at the Scottish Open, Andrew qualified for the 146th Open Championship at Royal Birkdale in which he had 3 rounds even and under par over the 4 days. A T44 finish for Andrew a great result for the Queenslander playing his first major.
- 2008: Playing his rookie year on the Asian Tour Dodt recorded six top-10 finishes, which included a runner-up result at the Vietnam Masters where he lost in a play-off;
- 2007: Dodt earned playing rights on the Asian Tour when he finished joint fifth at the Asian Tour Qualifying School.
- 2007: won the third annual Kel Nagle Plate for being the leading rookie professional at the Australian PGA Championship.
- 2015: Claimed his second European Tour victory at The True Thailand Classic.
- Qualified for the 146th Open Championship at Royal Birkdale in 2017