The former Wimbledon champion recorded a 7-6 6-2 6-3 victory on home soil in the decisive singles rubber.
Australia trailed their opponents 2-0 at one stage, but Hewitt's triumph in Darwin helped secure a dramatic turnaround to secure a place in the final four.
Great Britain and France will conclude their quarter-final affair later today, with the winners of that one to face Australia in the next round.
"We had to rally together and find a way to get the win," Hewitt told reporters after his triumph.
"I've always said some of my greatest wins are in Davis Cup and some of my toughest losses are in Davis Cup so I'm going to enjoy this one. This is what dreams are made of."
Argentina, Belgium, Serbia and Canada all still remain in the competition.