Britain were looking for a repeat of their victory over the same opposition in the last eight two years ago and made the perfect start when Dan Evans thumped Peter Gojowczyk 6-2 6-1.
But Cameron Norrie suffered his first defeat of the week, going down 7-6 (6) 3-6 6-2 to Jan-Lennard Struff, and Kevin Krawietz and Tim Puetz won a tense deciding doubles rubber 7-6 (10) 7-6 (5) against Joe Salisbury and Neal Skupski.
It is therefore Germany who will travel to Madrid for the semi-finals on Saturday while Britain head home and will almost certainly have to win a qualifying tie in early March to book their place in next year’s finals.
Smith said: “Everyone is gutted, obviously. We’ve had some good days, but it’s a tough day to swallow. Can’t fault any of the guys’ efforts or the team bench. Everyone gave their best effort.
“The sets could have gone either way and it could have been a different outcome. But it wasn’t to be today.”
This tie was once again played behind closed doors in Innsbruck but both teams provided vocal support, particularly during the doubles clash, which could scarcely have been tighter.
The match featured four of the world’s best exponents of the format, and there were some terrific rallies, but Salisbury and Skupski were left to rue missed opportunities after leading 6-4 in the first-set tie-break and 5-0 in the second.
Salisbury is a four-time grand slam champion but this week has been his first experience of Davis Cup tennis.
He said: “They played well at the end of the second-set tie-break, but we didn’t. We’re 5-0 up, we should win the tie-break, simple as that.
“I have loved this week, loved playing for Great Britain I think it’s extra pressure, which often I feel helps me to perform well and I feel like I have played my best in some of the biggest matches that I have played. But this week that didn’t happen.”
It was a similar story for Norrie, who has had an exceptional season but understandably looked to be running out of gas a little here.
He recovered from 1-4 to serve for the opening set but could not take advantage either of that opportunity or a 6-4 lead in the tie-break. Norrie appeared to have turned things around when he levelled the match but Struff finished with a run of five games in a row.
“I was really disappointed with my level,” said Norrie. “I think I didn’t serve as well as I did in the previous matches. He played some good tennis in some big moments, especially in the third there. The first set should definitely have been mine.
“Overall, the week was excellent. Really enjoyable way to end the season, and I’m so proud of everyone on the team. They left it all out there. We’re all gutted but I think, for the future, it’s going to be good to experience this and definitely going to make me and everyone else want it even more.”
The day had started so promisingly for Britain with the performance of Evans, who swatted aside Gojowczyk in just 55 minutes.
Germany captain Michael Kohlmann sprang a surprise by selecting Gojowczyk, who had not played a Davis Cup match since 2014 and was on a five-match losing streak, instead of Dominik Koepfer.
Having been expecting to face a left-hander, Evans instead found himself up against a right-hander, but Gojowczyk made a horribly nervous start and never recovered while his opponent got better and better.
Britain’s players will now take a well-earned holiday but it is only a month until the new season begins with a very similar competition, the ATP Cup in Australia.
Smith added: “We’ve got to look positively at some of the things we have done this week and improve on the stuff we can do even better next year.
“Now it’s time for some rest for these guys. They fully deserve it. We have lost today, but that happens. I think the most important thing for them is what they have achieved all throughout the year. They should be really, really proud.”