World number three Andy and older brother Jamie recovered from a set down to secure a thrilling 4-6, 6-3, 6-4, 6-7 (6/8), 6-4 victory in a gruelling four-hour rubber.
Victory gave the home side a 2-1 lead and it means Andy Murray can now clinch the tie for Great Britain in Sunday's first reverse singles as they attempt to win the competition for the first time since 1936.
The 28-year-old will secure a final showdown against either Belgium or Argentina in November if he defeats scheduled opponent Bernard Tomic.
If he loses, British world number 300 Dan Evans is due to face teenager Thanasi Kokkinakis in what would be the decisive second reverse singles.
"It was an incredible match and we had to come back from the disappointment of losing the fourth set," two-time Grand Slam winner Andy said.
"We stuck together like brothers should and managed to come up with enough good returns. It was as emotionally draining as much as it was physically."
Australia, 28-times winners of the tournament, had been looking to reach their first final since 2003 but are now on the brink of a first defeat against old rivals Britain since 1978.
"Davis Cup doubles over five sets can sometimes only rely on one or two points," said Hewitt, who is featuring in his final Davis Cup campaign.
"Doubles is always crucial but there is still two matches to go and we have to come up with the best plan to win those matches and give ourselves an opportunity to go through to the final."
With the tie delicately balanced after Friday's singles, British captain Leon Smith put Andy alongside Jamie instead of Dominic Inglot as he opted for the pairing who secured a four-set victory over Nicolas Mahut and Jo-Wilfried Tsonga in the quarter-final against France.
