Murray beat Sam Querrey 7-6 (7-5) 6-7 (3-7) 6-1 6-3 to give Britain an unassailable 3-1 lead in the best-of-five tie in San Diego.
It is the first time Britain have beaten the USA since 1935.
They will travel to Italy in April for the quarter-final, which will almost certainly be played on clay.
"This is a massive team effort," GB captain Leon Smith told BBC Sport.
"People don't see the work off the court. We should be proud of the team spirit and what the guys have achieved here.
"I was saying to the guys during the week that we've used 11 different players over the last few years - a true team effort. It's great to have this team spirit running through."
The victory was set up by a perfect opening day on Friday, when Murray beat Donald Young before James Ward upset Querrey over five sets.
Murray was rested from the doubles on Saturday, which saw Colin Fleming and Dom Inglot lose to the world number one Bryan brothers, but the Wimbledon champion completed the job over almost three hours against Querrey on Sunday.
"Sam came out playing extremely aggressive, he was very aggressive on my serve," said the Scot.
"Murray tried to set the tone from the very first
point with a stinging forehand return, but Querrey - playing with so
much more freedom than against James Ward on Friday - made life very
awkward in the first two sets. Murray had so much more variety from the
back of the court, though, and his sliced backhand threw Querrey off his
stride. This was a win that might not have been possible without Ward's
Friday heroics: it was a genuine team effort once again."
"I changed tactics at the beginning of the third set and I was able to dictate a lot of points after that."
Murray, 26, took a grip on the match after the first
two were shared in tie-breaks, and overcame a late scare when he had to
recover from 15-40 down to serve out the match.
"You have a responsibility to your team-mates to play well," said Murray, "but I also have a lot of experience in the Davis Cup and the Slams, so you know how to deal with it relatively well."
Querrey, ranked 49th, put up a strong fight following his surprise loss to Ward, twice coming back from a break down in the first set and edging the second.
But with the American crowd sensing an unexpected chance to get back in the tie, Murray slammed the door shut as he won the first four games of the third set and got the decisive break in game six of the fourth.
It was Murray's 17th successive Davis Cup singles win and secured an eighth win from nine matches for Smith, who took over the captaincy in 2010 when Britain were on the verge of relegation to the lowest Europe/Africa Zone Group III tier.
They now face the prospect of a quarter-final on 4-6 April against Italy.
The Italians, led by world number 15 Fabio Fognini and number 31 Andreas Seppi, beat Argentina 3-1 in Mar del Plata.
"This is the first time I've been involved in Davis Cup that we've had a chance of winning the competition," said Murray.
"It's good for me to play on the clay. Often going into the clay season I haven't played any matches on it for 11 months. It'll be a tough match."
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