Late tries from Dougie Hall and Stuart Hogg had brought the Warriors to within touching distance of upsetting the Heineken Cup finalists but Joe Schmidt's men held on to deny his opposite number a fairytale ending.
However despite the result Lineen was adamant the Warriors could take pride in their performance and will be back challenging for the major prizes again next year.
"For Glasgow it is onwards and upwards," said Lineen. "I'm just so proud of what they've done this season.
"As I said in the dressing room, Glasgow are going to win something. We're edging closer.
"They battled and they never gave an inch but we were just maybe not quite as slick as them. We didn't have that composure and direction to get over the line. These are the little things and we're not far away.
"We had three squint line-outs, which is pressure. Duncan Weir misses a couple of kicks, which he wouldn't normally miss. That's where the pressure comes on individuals.
"But these guys will learn. Duncan's been fantastic for us. On these occasions, you need everyone really performing.
"Our defence has been fantastic all season. But you can't just defend the whole game. There are parts of our game we need to really work on and I'm sure Gregor will do that. That's his forte in attack.
"We've blooded in some new young players. There are some other guys coming in to look at the ball carrying in the back row. We're strengthening the team. It's looking good for Glasgow.
"It's someone else's turn now, I was lucky enough to get an opportunity nine years ago and now I'm lucky enough to still be involved in Scottish rugby in another role. That's fantastic for me."
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