Munster march into play-offs
Saili, who replaced Jaco Taute in the second half, came up trumps with the game's only score in the final 10 minutes as the visitors made the most of a yellow card for Stuart Hogg.
A tight match saw the boot dominate, but when it mattered most Saili's effort proved the difference between the teams.
A man light late on, Glasgow pushed to move back ahead, but Finn Russell opted against a drop-goal shot and his team went down to defeat.
With both sides in excellent form and Munster on the back of thrashing Racing 92 last weekend, an eagerly-anticipated encounter was on the cards.
And at a packed Scotstoun, the game didn't disappoint.
A tight first period ended level at 6-all, the two sets of forwards going hell for leather at each other on a chilly Scottish night.
Glasgow flyhalf Russell, so impressive in recent weeks, swapped penalties with opposite number Tyler Bleyendaal in an opening quarter which saw physicality and defensive organisation dominate proceedings.
Munster initially looked the more dangerous in attack, but after failing to make the most of a number of penalties, they slipped behind.
Lions prospect Hogg nailed a long-range penalty to put Glasgow 6-3 up, but after Bleyendaal had booted another kick and Hogg missed with a second monster effort, the two teams went into the break level.
It had been nip and tuck though there remained a feeling Glasgow were yet to fully find the attacking form which saw them do the double over Racing.
The start of the second period saw Russell and Bleyendaal then exchange further kicks at goal as the second 40 went a similar way to the first.
Conor Murray's sublime kicking game ensured Munster kept things tight defensively, a big line-out steal in the 54th minute summing up their effort.
For all their resilience, though, the Irish side struggled in their search for a try and Russell's third penalty of the evening made it 12-9 on the hour mark.
A raft of replacements were then introduced, but the key moment arrived with 10 minutes remaining.
A Munster attack saw Andrew Conway put into space and the right wing looked like he was going to make it to the line.
He was scragged round the neck by Hogg and although the decision not to award a penalty try was the right one, the fullback correctly saw yellow.
Munster immediately took advantage and Saili's try came less than a minute after Hogg walked from the field.
That, in the end, proved to be the winning moment.
The scorers:
For Glasgow Warriors:
Pens: Russell 3, Hogg
For Munster:
Try: Saili
Pens: Bleyendaal 3
Yellow card: Stuart Hogg (Glasgow Warriors, 70 – foul play, head-high tackle)
Teams:
Glasgow Warriors: 15 Stuart Hogg, 14 Tommy Seymour, 13 Mark Bennett, 12 Alex Dunbar, 11 Lee Jones, 10 Finn Russell, 9 Ali Price, 8 Josh Strauss, 7 Ryan Wilson, 6 Rob Harley, 5 Jonny Gray (captain), 4 Tim Swinson, 3 Zander Fagerson, 2 Fraser Brown, 1 Gordon Reid.
Replacements: 16 Pat MacArthur, 17 Alex Allan, 18 D'Arcy Rae, 19 Matt Fagerson, 20 Chris Fusaro, 21 Grayson Hart, 22 Nick Grigg, 23 Peter Murchie.
Munster: 15 Simon Zebo, 14 Andrew Conway, 13 Jaco Taute, 12 Rory Scannell, 11 Keith Earls, 10 Tyler Bleyendaal, 9 Conor Murray, 8 CJ Stander, 7 Jack O'Donoghue, 6 Peter O'Mahony (captain), 5 Donnacha Ryan, 4 Jean Kleyn, 3 John Ryan, 2 Niall Scannell, 1 Dave Kilcoyne.
Replacements: 16 Rhys Marshall, 17 James Cronin, 18 Thomas Du Toit, 19 Dave Foley, 20 Billy Holland, 21 Duncan Williams, 22 Ian Keatley, 23 Francis Saili.
Referee: Luke Pearce (England)
Assistant referees: Matthew Carley (England), Wayne Falla (England)
TMO: Graham Hughes (England)
Source: @ChampionsCup
