The only problem is that their opponents this evening in Dublin are three-time champions Leinster, whose leading icon, Brian O'Driscoll, is playing his last-ever game. Here are five areas that will be key in deciding the outcome.
1 The Set-Piece
The Glasgow front row has never been embarrassed this season, and Gordon Reid has grown in stature as a scrimmaging prop. However, they are up against a serious scrummaging machine, for the Leinster pack destroyed their Ulster opponents at some early scrums in their semi-final two weeks ago. The two sides look pretty even in the lineout, too, where Leinster have the giant that is Devin Toner, but Glasgow may have more catching options.
2 The Playmakers
Finn Russell, below, has been a revelation for Glasgow in the final two months of the season, playing with a maturity far beyond his 21 tender years. He has slotted in beautifully for Glasgow at 10 or 12, with a game that is as well balanced as his outlook. However, he is up against one of the great fly-half grafters in the shape of Jimmy Gopperth, who was signed from Newcastle as a backup player a year ago but has emerged as Leinster's first choice in the position.
3 The Strike Runners
Glasgow couldn't buy tries earlier in the season, but their passage to the semi-final was helped by the clutch of bonus points they picked up in their last few regular-season games. Aside from the flying machine that is Sean Maitland, Alex Dunbar can carve through gaps at will and Tommy Seymour is a great opportunist - as his hat-trick against Treviso showed. Yet Leinster have one of the all-time best game-breakers in Brian O'Driscoll, while Zane Kirchner, Fergus McFadden and Rob Kearney, left, are a pretty handy back three.
4 The Breakdown
As Gregor Townsend has pointed out, Leinster pretty much wrote the book on what modern breakdown play is all about. Fierce and committed in contact, every player from 1 to 15 has been schooled in speeding up his side's possession and slowing things down for the opposition. Against that, their advantage has diminished in an area where other sides have worked hard to catch up. In Rob Harley, Glasgow have a player who will scrap for everything, while Josh Strauss is a beast on the ball.
5 The Mental Game
Leinster have all the big game experience that Glasgow lack, and that has to be a strength in a winner-takes-all contest. In terms of rugby finals, they've been there, done that and built up an impressive collection of winners' T-shirts. Glasgow are in new territory today, but their momentum and the confidence engendered by nine straight wins cannot be discounted. More than ever before, Glasgow now believe they can do it.