David Nalbandian secured Argentina's 3-2 passage against Sweden by defeating Andreas Vinciguerra in the decider.
BBC: David Ferrer secures Spain Davis Cup quarter-final spot
But Nalbandian insisted that he should not have been forced to default the match.
"We passed very tough moments in the last year-and-a-half with the surgery, " Nalbandian said.
Murray next plays Argentine David Nalbandian as he bids for a record third successive Masters title.
Nalbandian extended his brilliant run of form by crushing local favourite Gasquet in just over an hour.
Former world number three David Nalbandian, now ranked 24, has hired Argentine Martin Jaite as his coach.
Their match had followed a classic encounter between fourth seed Andy Murray and Argentina veteran David Nalbandian.
Gaudio awaits the winner of Wednesday's second quarter-final between fellow Argentine David Nalbandian and Brazil's Gustavo Kuerten.
Nalbandian becomes the first player ranked outside the top 100 to win an ATP Tour title this year.
"The key to winning the match was my better serve and playing high on his backhand, " said Nalbandian.
Nalbandian could make it three Argentines in the semi-finals, after Guillermo Coria's win over Carlos Moya on Tuesday.
Schuettler, seeded eight, will play David Nalbandian in the semi-finals after the Argentine beat Feliciano Lopez 6-4 6-4.
Tenth seed David Nalbandian made an early exit, beaten 4-6 6-4 7-5 by fellow Argentine Juan Ignacio Chela .
But after a break of serve apiece, Nalbandian dominated the tie-break and took it with a sweeping forehand pass.
Massu will play David Nalbandian in the semi-finals after the Argentine beat Germany's Rainer Schuettler 4-6 7-6 (7-5) 6-3.
Another Frenchman, Fabrice Santoro, also reached the last eight with a 6-0 6-2 defeat of Argentina's David Nalbandian .
Lopez and Fernando Verdasco will face Nalbandian and Agustin Calleri in Saturday's doubles.
Tournament director Chris Kermode told BBC Sport that officials had been left with no other option than to default Nalbandian.
The Scot was below his best in beating Fernando Verdasco on Thursday and came up short against a rejuvenated Nalbandian.
The Spaniard has beaten Federer four times this season but Federer is refusing to look beyond his opener with Nalbandian.
Nalbandian's win over Ljubicic was sweet revenge following his defeat to the Croatian in the semi-final of October's Madrid Masters.
The final offers Nalbandian the chance to atone for his three-set defeat to Federer in the group stages last weekend.
David Nalbandian and Eduardo Schwank earned Argentina a 2-1 lead on Saturday after an epic doubles victory in Buenos Aires.
With Cypriot eighth seed Baghdatis landing only 39% of his first serves in the opening set, Nalbandian took it with ease.
Nalbandian, who has been without a coach since last year, and Safin both went out in the third round of Wimbledon.
The win put Nalbandian, a runner up in Madrid in 2004, into his first semi-final of the year after a disappointing season.
American 16th seed John Isner survived another marathon match as he defeated 2006 semifinalist David Nalbandian 4-6 6-3 2-6 7-6 (7-5) 10-8.
Argentina, who thrashed Germany 5-0 in the first round, have David Nalbandian, Gaston Gaudio, Mariano Zabaleto and Lucas Arnold in their team.
As a result Nalbandian, who had been leading sixth-seeded Cilic of Croatia 7-6 (3), 3-4 when the incident occurred, defaulted the match.
Murray, who missed Madrid through injury, breezed through opener 6-1 but Nalbandian hit back to level by taking the second set 6-4.
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