Murray secured a three-set win over Bernard Tomic to give his team an unassailable 3-1 lead in the tie and earn a title match against Belgium, who later completed a 3-2 win over Argentina.
His doubles victory with brother Jamie Saturday had proved the pivotal moment as they saw off Lleyton Hewitt and Sam Groth in a five set thriller.
It had led to worries that Murray might be fatigued, but he comfortably beat Tomic 7-5 6-3 6-2 in one hour and 46 minutes.
The victory completed a remarkable comeback for the GB team, which nearly dropped out of the European/African zone in 2010 but has since moved up to the World Group with growing success.
Much of that is down to two-time grand slam champion Murray, who has returned to play regularly for his Davis Cup team and boasts a singles record of 25 wins against just two defeats in the premier team competition in tennis.
It was never seriously under threat from Tomic, who did fight back from a break down in the first set, before conceding his own service again.
There was speculation that veteran Hewitt, in his final full season on the ATP Tour, would be drafted in for the reverse singles, but either would have struggled against Scot Murray, who was showing inspired form before his home fans at the Emirates Arena.
A single break of service gave him the second set and as Tomic tired he was broken two more times in the third as a triumphant Murray wrapped up victory.
In Brussels, the home side fought back from 1-2 down overnight to win the two reverse singles matches and set up a clash with Britain in November.
First David Goffin leveled the tie when he beat Diego Schwartzman 6-3 6-2 6-1 to set up a decider between home player Steve Darcis and Federico Delbonis.
Darcis took the opener but Delbonis hit back by dominating the second set. The third proved pivotal as Darcis took it 7-5 and he reserved his best play until the fourth set tiebreaker, claiming victory on the second match point.
It set up a repeat of the 1904 Davis Cup final, won 5-0 by Britain, but this time Belgium will have home advantage from November 27-29.