That put the hosts 2-0 up after James Ward's hard-earned win earlier, and victory for Leon Smith's team in the doubles on Saturday will ensure they return to the second tier of the competition for the first time since 2009.
Kiss is a full-time law student who plays professional tennis so infrequently he no longer has a world ranking, and the only real question appeared to be whether he could do better than his equivalent in the Luxembourg team had managed during Britain's last tie in July.
On that occasion Murray had beaten poor Laurent Bram without dropping a game, losing only 15 points in the entire match, but this match turned out to be rather different.
The same scoreline looked on the cards after a one-sided first set, but Kiss proved himself to be, if not in the same class as the world number four, a decent striker of a tennis ball and someone who relished the opportunity.
He stopped the rot in the eighth game and almost unbelievably broke Murray's serve for the first time when he served for the match at 5-3 in the third set.
The Scot then proceeded to let three match points slip away as a Friday evening stroll took an unexpected turn before coming through the tie-break, clinching victory on his fifth chance.
Earlier, an ailing Ward had beaten Hungarian number one Attila Balazs 6-4 6-4 4-6 6-4 in a survival of the fittest encounter that lasted more than three hours.
Ward had to leave the court to be sick during the second set and battled cramp for more than half the match before coming from behind in the fourth as Balazs also began to cramp.
"I didn't want to throw up all over the court so I went outside the door," said the 24-year-old of the moment when he came out to serve at 5-4 in the second set.
"I've never been sick (during a match) before so it was obviously strange but they said I had to come straight back on and play or I'd forfeit.
"I'm playing for my country. You've got to keep going and you've got to keep fighting, even though it was difficult.
"It's the first time I've ever cramped on a tennis court so it was a weird feeling for me and I didn't really know what to do. I don't even know why it's come on. I think it's tension and wanting to win so much."
Ward, who lost the opening rubber to Gilles Muller of Luxembourg last time out, put his physical problems mostly down to nerves.
He said: "I'm ranked a lot higher than the other guy and it sometimes gets to you on the big occasion.
"I lost the last one against Muller and I was a bit disappointed with my performance there. Since then I've had a good summer and it's a good opportunity to show everyone how well I've been playing and what I want to do for the future.
"I've got my family here as well. It's the first time a lot of them have seen me play for a while so it was a big occasion for me."
Captain Smith said: "Andy did his job really well and after what James went through he did a great job getting through it.
"It's 2-0 and we've got an excellent doubles team but you can see what happens, even the guy that played Andy, he's not ranked and he's swinging away and you have to guard against that.
"Ross and Colin are very professional and they'll go out very focused and will look to do a good job."