Despite losing the doubles on Saturday, Britain had began the deciding day 2-1 ahead in the best-of-five contest because Querrey had lost a five-set epic to world number 175 James Ward on Friday after Murray had overcome Donald Young 6-1 6-2 6-3.
"We have a really good team spirit," Murray told BBC Sport. "I've known a lot of the guys for a long time.
"There's pressure because you have responsibility for your team-mates but at the same time they've put me in a great position and have won important ties themselves.
"Sam came out playing extremely aggressive, he was very aggressive on my serve. I changed tactics at the beginning of the third set and I was able to dictate a lot of points after that."
Leon Smith's side will now travel to Italy in April, with the winners to play either Switzerland or Kazakhstan in the semi-finals.