In Brussels, Argentina go into the final day leading Belgium by the same margin in their semi-final after Leonardo Mayer and Carlos Berlocq beat Steve Darcis and Ruben Bemelmans 6-2 7-6 (2) 5-7 7-6 (5) in a four hour marathon.
The Murrays kept their vociferous home supporters on the edge of their seats at Glasgow's Emirates Arena before triumphing 4-6 6-3 6-4 6-7 (6-8) 6-4.
"It was an incredible match, to come back from the disappointment of losing the fourth set," Andy Murray, ranked third in the world, told the BBC.
"We kept creating chances, we stuck together like brothers should and managed to come up with enough good returns."
Older brother Jamie, the doubles specialist, said: "We let it slip a bit but we kept fighting and got right back on it. It was so good, it was so noisy, it was magic.
"Scottish people don't get the chance to see Andy play very much and when they do they let him know it."
Andy Murray won the opening singles on Friday 6-3 6-0 6-3 against Thanasi Kokkinakis and plays Australian No.1 Bernard Tomic on Sunday's deciding day.
Tomic, who beat Dan Evans on Friday to square the tie, lies 20 places below Murray in the world rankings.
While Argentina lead into the reverse singles, Belgian number one David Goffin will take heart from Meyer having gone through two tiring matches in two days -- even if the Argentine sounded confident.
"It's been four hours, two days, long matches but I have energy," said Meyer, who beat Darcis in four sets with two tie-breaks on Friday.
"Davis Cup is like this, and I love playing in Davis Cup."
(Reporting by Clare Lovell, editing by Alan Baldwin/Greg Stutchbury)
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