After a promising start which saw Mike Tindall go over for the first try, yellow cards for front rowers Dario Chistolini and Koree Britton cost Gloucester dear.
Two tries from young winger Teddy Thomas, and a score off the back of a rolling maul for hooker Arnaud Heguy saw the visitors lead 27-13 at the break.
Former Gloucester player Dimitri Yachvili orchestrated proceedings for the Basque side and he put Takudza Ngwenya away to feed Benoit Baby for the fourth try.
Replacement Seremaia Burotu added a fifth although Gloucester showed a lot of fight to cross twice late on, first through Shane Monahan and then hooker Huia Edmonds.
And despite the disappointment of being knocked out of Europe having gone through the group stages with a 100 percent record, Davies is confident his team will build on the experience.
He said: "It was a strange old game. We played our part in it and it was a great learning experience for this group of players.
"I think it's the first time in four seasons that Gloucester have got to a European quarter-final. Those players will have learnt that you really have to be on your mettle in games like this.
"We couldn't back up the performance against Harlequins from last week. We just lacked a little bit of intensity.
"There was a lot of effort but that's where this group is at the moment - it's learning.
"The yellow cards put a lot of pressure on us and we also conceded points after we had scored.
"We played a lot of rugby but Biarritz showed that they are a very clinical side. Their driving lineout was very powerful and very strong and their other points came from broken play.
"There were a couple of key moments. If we'd have stopped their driving maul they wouldn't have scored two of their tries. If we'd have scored before they did with that break from Sione Kalamafoni it might have been a different story. If we hadn't had two yellow cards, who knows?"
Tries: Tindall, Monahan, Edmonds
Conversions: Burns 2
Penalties: Burns 4
Tries: Thomas 2, Heguy, Baby, Burotu
Conversions: Yachvili 5
Drop Goals: Traille