Cheered on by a partisan crowd, the former Wimbledon champion survived some early scares against the big-serving American before taking charge to lead Britain into a 3-1 lead.
In the quarter-finals they will face a home tie with France after they beat Germany with a day to spare.
"It's been incredible the whole week, to be in the quarter-finals for the second time in a row is a credit to everyone," world No 5 Murray, revelling in playing in front of a Scottish crowd, said.
"Everyone was thinking it was going to be comfortable against John because he would be tired but that put more pressure on me and I definitely felt that."
Isner, beaten in five sets by British No 2 James Ward on Friday, had three set points in the opener but could not convert and despite some fierce resistance in the third set he could not keep the 32-times champions alive.
Italy, who beat Britain in the quarter-finals last year, went down to Kazakhstan in Astana, where Aleksandr Nedovyesov, ranked 130th in the world, stunned world No 22 Fabio Fognini in five sets to seal a 3-2 victory.
Kazakhstan will travel to former Davis Cup heavyweights Australia who enjoyed a 3-1 victory over 2012 and 2013 winners Czech Republic in Ostrava -- Bernard Tomic clinching the deal with an impressive 7-6(4) 6-3 7-6(5) win over Lukas Rosol on Sunday.
Australia have reached the last eight for the first time since 2006.
World number one Novak Djokovic's Serbia crushed Croatia 5-0 and will take on the winners of the match between Argentina and Brazil in Buenos Aires.
Holders Switzerland, without Roger Federer and Stanislas Wawrinka, were locked at 2-2 with Belgium in Liege.
