Early in the second half, Glasgow only trailed by one point but a lack of composure cost them with two of the three tries they conceded coming from intercepts.
The set piece was also a problem area as the Warriors lost six lineouts on their own throw.
And with champions Munster coming to Firhill next in the RaboDirect Pro12, Lineen is confident his side are on the right track once they address those failings.
He said: "No one thought we would have a chance over here but deep down the boys had a lot of belief with the way with the way we trained and played.
"But ultimately an area that's usually our strength, the lineout, just did not function. "To lose six is disappointing and he boys are better than that. We did not really have that platform but we went in 15-11 down at halftime.
"The second half we got to 15-14 and we were there. But then just a bit of composure and their senior guys came to the fore playing in the right areas.
"Guys just need to look at the little mistakes. When you look at the bigger picture there were some really good performances.
"We will look back on the game and two tries were from intercepts and one was from a 5m scrum.
"It is just making sure we do the basics well and go back to the drawing board for the lineout and I have no doubt that we will sort that out."
There were plenty of new faces in the Glasgow ranks with prop Michael Cusack and full-back Stuart Hogg making their debuts along with try-scorer Troy Nathan.
But it was the performances from young fly-half Duncan Weir and experienced centre Rob Dewey that delighted Lineen the most.
He said: "We will learn a lot from that. There was some really good stuff from Duncan Weir, Rob Dewey in the centre was outstanding and Stuart Hogg at full-back had a really good debut.
"Michael Cusack did well in the scrums. Tom Ryder always gives us his all and Rob Harley as captain had an outstanding day.
"Troy he did a lot of good things. He is just what we need, helping Rob outside him and also inside him with Duncan.
"Some of [Duncan's] long-range kicking was just world-class. He did some outstanding things and it is great to work with him. He is just going to get better and better."
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