In a game dominated by kicking and the tight exchanges, Lineen's side showed great tenacity at the breakdown to disrupt Biarritz's flow.
Julien Dupuy kicked two penalties for the visitors but they were unable to find a winning score.
Lineen said: "My guys displayed an unbelievable amount of guts to get this result.
"In the changing room at the interval, I looked at them all - and their eyes told me the depth of their self-belief and that they would win.
"Biarritz made a lot of mistakes, but most of them were forced on them by our pressure.
"There is no doubt that Biarritz will give us a very warm reception when we go there on Friday - but we won't be making the journey for anything other than another win.
"I am really proud of the way the guys dug in and it was without doubt the best result since I came to the club.
"Our defence against one of the best teams in the tournament was outstanding and our scrum was quite brilliant."
Biarritz backs coach Jack Isaac admitted: "We did not deserve to win and it would have been a shame for Glasgow if we had.
"We are all very disappointed because it was the worst we have played for some time. I was surprised our boys were not up for it.
"We only came alive in the last few minutes and we got what we deserved.
"We have a lot of work to do before we face Glasgow again at home."