The Wimbledon champion brought his team level at 1-1 when he swept aside Andreas Seppi 6-4, 7-5, 6-3 in a match continued from day one.
He then teamed up with fellow Scot Colin Fleming in the doubles to produce a 6-3, 6-2, 3-6, 7-5 victory over Simone Bolelli and Fabio Fognini.
Their success gave Britain a 2-1 lead going into today's reverse singles, following James Ward's defeat to Fognini in Friday's opening rubber, and means Murray will take Great Britain to a first Davis Cup semi-final in 33 years if he beats Fognini today.
"This is why we are so blessed to have Andy in the team," said Great Britain captain Leon Smith. "He's a fighter, he's a champion, and we have seen over the last couple of days how much he brings to the team. His Davis Cup record is pretty amazing."
On Thursday there had been doubt as to whether Murray would take to the court at all as he battled a virus, but he restarted his match against Seppi one set up and level at 5-5 in the second and wasted little time in strengthening his hold on the contest.
He took the first two games of the day to move 2-0 up in sets before his opponent threatened to stage a comeback when he broke for a 3-1 lead in the third.
Murray, though, hit back, rattling off five straight games to seal victory in three hours and seven minutes.
The world No 8 returned for the doubles and the decisive break for the British pair in the first set came in the eighth game after Fognini double faulted, then smashed long.
Britain broke in the third game of the second set following a Bolelli error and again in the seventh when Murray's return down the line was too good for Fognini. However, Italy broke in the fourth game of the third set and saw it out to reduce the deficit to 2-1. The fourth set was a see-saw affair, with Britain securing the fifth break of serve - and the match - by taking the third break point available. Fognini takes on Murray in the first of today's singles match, with Seppi due to play Ward in what could be a dead rubber.
In other ties, the Czech Republic stayed on course for a third straight Davis Cup title by booking their semi-final spot, but Switzerland are on the brink of a shock defeat to Kazakhstan.
The "dream team" of Stanislas Wawrinka and Roger Federer were expected to ease to victory in Geneva but both must win their singles matches today after they lost the doubles to Andrey Golubev and Aleksandr Nedovyesov to trail 2-1. Golubev had stunned Wawrinka on Friday before Federer drew the hosts level by beating Mikhail Kukushkin.
Wawrinka seems to be feeling the pressure that comes with trying to live up to being a Grand Slam champion and he was well below his best again as Golubev and Nedovysov triumphed 6-4, 7-6 (7/5), 4-6, 7-6 (8/6).
Federer insisted Switzerland are still confident of coming through, saying: "I'm not overly worried about what happened the last couple of days.
"We knew that Kazakhstan have been playing really well in Davis Cup. It's up to us now to shine and prove to ourselves that we did take the right decisions, that we are good players.
"We are still the favourites for both singles matches but we have no more margin for error, and maybe that's going to make us play even better. I'm actually very excited."
Wawrinka will be first in action against Kukushkin before Federer takes on the impressive Golubev. The Czechs moved into an unassailable 3-0 lead over Japan when Radek Stepanek and Lukas Rosol, who both won their singles matches on Friday, combined for a 6-4, 6-4, 6-4 victory against Tatsuma Ito and Yasutaka Uchiyama.
Meanwhile, France kept their hopes alive against Germany when Julien Benneteau and Michael Llodra defeated Andre Begemann and Tobias Kamke 6-1, 7-6, (7/5), 4-6, 7-5.
The hosts endured a miserable first day in Nancy, with Benneteau beaten by Kamke before Peter Gojowczyk stunned Jo-Wilfried Tsonga.