A draw would mathematically guarantee the Warriors passage to the play-offs, but players and coaches alike are gunning for the win that could see them climb into third place come the final whistle on a night when the club will bid farewell to Firhill ahead of the summer move to Scotstoun Stadium.
A measure of how difficult a challenge awaits Glasgow tomorrow evening is contained in the sobering statistic that they haven't beaten Connacht in the last four meetings between the teams.
Two of those games have ended in draws, most recently at the Sportsground in Galway back in February, with Connacht doing the double in last season's head-to-heads.
Eric Elwood's team are looking to secure four straight RaboDirect PRO12 wins for the first time in their history, and are determined to force a best-ever placing in the competition table.
Warriors head coach Lineen believes the match will be up there with the toughest his men have faced, telling www.glasgowwarriors.org: "There's been a lot of talk outside the camp about the various play-off scenarios, but our message to the players has been very simple.
"This will be a hugely difficult game against tenacious and confident opponents, and our minds will be focussed on one thing and one thing only: doing whatever it takes to win the match.
"Collectively, we've worked massively hard all year to give ourselves this opportunity. It's about everyone concerned bringing the same intensity, accuracy and spirit they have done throughout the campaign and making sure we earn the right to prolong our season."
The Warriors starting line-up for the game shows seven changes from that which ran out in the 13-8 win away to Benetton Treviso two weeks ago.
Three of the changes come in the backline, where Henry Pyrgos is rewarded for his fine recent form with a start at scrum-half, Federico Aramburu replaces Tommy Seymour on one wing, and DTH van der Merwe is selected on the other.
The Canada international, who emerged from the bench in Treviso to score the winning try, will be making his first start of the season, having recovered from a shoulder injury.
Aramburu, meanwhile, is to captain the team, with regular skipper Al Kellock among the substitutes.
There is an all-new front row, where Jon Welsh, Dougie Hall and Moray Low team up, while Tom Ryder, one of the Warriors' most consistent performers this season, comes into the boilerhouse.
Ryder and Welsh were yesterday named in the RaboDirect PRO12 Dream Team for season 2011/12, along with stand-off Duncan Weir.
Lineen explained: "We've placed huge emphasis all season on the importance of a 23-man effort, and that's absolutely the case again this week.
"There are a few bumps and bruises floating around as you'd expect at this stage in a long, hard year, but it's also right and proper that we reward players for what they've produced both in the last few games and over the piece.
"Henry Pyrgos and Tom Ryder have both been excellent for us, and they'll be a big part of what we absolutely believe is the right team to win the game tomorrow.
"In terms of the front row, we're in the fortunate position of having nine or more guys who can all step up and do the job we need them to. Five of the starting pack have international experience, and the other three have been real standouts for us all year.
"In the backline, we're delighted to be in a position to give DTH a start. He underlined his class and game-changing ability against Treviso, and he's desperate to get back out there in front of the Warriors crowd and show them what they've been missing.
"Federico Aramburu is a worthy captain for the team. Because he hasn't always played and he's not necessarily got the highest profile, people outside the group perhaps underestimate what a key influence he is within the squad. Every single player here holds him in great esteem for what he's achieved in the game and how he conducts himself.
"The bench is the strongest I've ever named as head coach of Glasgow Warriors, and I'm expecting them all to have a part to play in helping us land the right result."
As well as being the last Warriors match at Firhill, tomorrow's game is a final home game for Lineen and defence coach Gary Mercer, along with a number of players.
Substitutes Calum Forrester and Colin Shaw are both moving on at the end of the season, as are starters Richie Gray and Aramburu. The 22-cap Argentina international has announced he will retire come the summer.
Lineen said: "It's great for myself and Gary and this group of players to have the chance to show our appreciation for the fantastic backing the Firhill crowd have given us down the years.
"The best way to do that is by winning the game and giving them a semi-final to look forward to, so we're taking sentiment out of it and keeping our minds on that specific task.
"I'm looking forward to seeing Firhill absolutely buzzing one last time. With players, coaches, staff and supporters all pulling together, we're more than capable of giving the ground a fitting send-off with a win."
Tickets to the vital clash (kick-off 7.30pm) are shifting fast - we've already registered our biggest ever pre-sale for a league match outside of the 1872 Cup derby against Edinburgh Rugby.
If you still need to book your spot, you can do so via www.glasgowwarriors.org, by calling 0844 800 3490 or in person at the Gordon Street and Sauchiehall Street branches of Greaves Sports our official retail partner.
Book before 5pm on Friday online or over the phone to save £5 on match-night adult prices.
The Ticket Centre at Firhill will be open from 6pm on Saturday, as will the turnstiles. We're expecting it to be very busy. All supporters are strongly advised to come early, especially those who still need to collect or buy tickets.
Fans planning to make a Super Saturday of it by enjoying Day 1 of the Emirates Airline Glasgow 7s at the Warriors' new home at Scotstoun can book a spot on the buses that will be boarding at 6.20pm to take supporters to Firhill.
Numbers are strictly limited, and tickets must be pre-booked. Head to www.glasgowwarriors.org or call 0844 800 3490 to book.
GLASGOW WARRIORS TEAM TO PLAY CONNACHT AT FIRHILL IN THE RABODIRECT PRO12 AT FIRHILL ON SATURDAY 5 MAY (KICK-OFF 7.30PM)
15 Stuart Hogg
14 Federico Aramburu (CAPTAIN)
13 Alex Dunbar
12 Graeme Morrison
11 DTH van der Merwe
10 Duncan Weir
9 Henry Pyrgos
1 Jon Welsh
2 Dougie Hall
3 Moray Low
4 Tom Ryder
5 Richie Gray
6 Rob Harley
7 Chris Fusaro
8 John Barclay
SUBSTITUTES
16 Pat MacArthur
17 Mike Cusack
18 Ryan Grant
19 Al Kellock
20 Calum Forrester
21 Chris Cusiter
22 Ruaridh Jackson
23 Colin Shaw
Not considered for selection due to injury: Rob Dewey (shoulder), Rory Lamont (leg), Fergus Thomson (shoulder), Ryan Wilson (knee).
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