Tommy Seymour, Chris Fusaro and Peter Murchie tries set up a frantic finale as the Warriors desperately searched for a late fourth try, but found the RaboDirect PRO12's bottom side a tough nut to crack.
Despite the disappointment of missing out on a five-point victory, and Ospreys' shock 23-22 win over Leinster, Lineen was quick to point out a top four place was still within their grasp.
He told www.glasgowwarriors.org: "Everyone is talking about the top four but we want to make it much more clear-cut.
"Our destiny is in our own hands and if we keep winning we can get into the top two, which would earn us a home tie.
"This is another really good result from us, the product of an excellent team display. The will to win of this group is second to none, and we've put ourselves in a good position to kick on and achieve something substantial from the season.
"The players, to a man, were outstanding tonight, and the same goes for the crowd. We'll be needing them all on top form next week for what will be another huge test against Cardiff."
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