The Americans had needed the victory to remain in the quarter-final tie after Jordan Thompson stunned Sock 6-3 3-6 7-6(4) 6-4 and Nick Kyrgios eased past John Isner 7-5 7-6(5) 7-6(5) in the opening singles on Friday.
Visiting captain Jim Courier gambled on bringing Sock in for Sam Querrey even though the U.S. number one had racked up 66 unforced errors in his loss to world number 79 Thompson.
Despite a rocky start, the gamble paid off when Sock and Johnson prevailed in a rollercoaster ride of a match played out on a packed and partisan Pat Rafter Arena.
Johnson sealed the deal when he lashed a huge forehand into the back of the court after two and a half hours of action to bring the Americans back into the tie at 2-1.
"Jordan played great yesterday, deserved the win and was the better player, but today we had to regroup and come out and get the win," said world number 15 Sock, who will face the in-form Kyrgios in the opening reverse singles match on Sunday.
It looked like Sock was still deflated by Friday's defeat when Australia raced to a 4-1 lead in the first set and Peers slammed down an ace to seal it in 27 minutes.
The Americans stormed back to claim a mirror-image second set, however, and broke the Australians twice to take a 3-0 lead and run away with the third.
The hosts got the crowd back into the game when a rasping Groth forehand return down the line gave Australia a decisive break for 4-2 in the fourth but the Americans would not be denied and muscled up to claim the decider.
The two countries have 60 Davis Cup titles between them but Australia have not won since they bagged their 28th crown in 2003, while the United States snared their 32nd a decade ago.
The prize for victory in Brisbane is a semi-final against Belgium or Italy. The Belgians led the Italians 2-0 after the opening singles in Charleroi.
"We're still alive," said Courier. "Our guys played an electrifying brand of doubles out there today and they had to because Sam and John are a tough combination.
"It's fun to have a little bit of a breather and to have something to play for tomorrow."
(Reporting by Nick Mulvenney; Editing by Peter Rutherford)
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