Fly-half Glen Jackson scored all bar five of Saracens' points
Glasgow (9) 17
Pens: Parks 4
Saracens (18) 21
Tries: Haughton, Jackson
Pens: Jackson 3
Saracens reached their first ever Heineken Cup quarter-final as they ended Glasgow's remote chances.
Richard Haughton's early try and two penalties from Glen Jackson, to three Dan Parks penalties, gave them an 11-9 lead as half-time loomed.
Jackson claimed a cute interception try to send the visitors nine points clear.
Bernardo Stortoni crossed to cut the gap to four points before Jackson and Parks exchanged kicks as Saracens claimed a home tie in the last eight.
Despite the heavy rain in the west of Scotland over the previous two days the match kicked off in dry conditions, although the swirling wind made conditions tricky.
The hosts knew they needed a bonus-point win to have any chance of going through and they were the first to threaten.
Hefin O'Hare nearly profited when Haughton dallied hoping a kick would go out, but after racing onto the bouncing ball the former rugby league man just put a foot in touch and was rightly called back.
The incident seemed to wake Haughton up and when Jackson drilled a low kick into the Glasgow touch-in-goal area O'Hare dithered as he waited for it to run dead.
Haughton used his express pace to hunt down the ball and the Television Match Official ruled he had dived to touch down millimetres inside the line.
Parks missed the conversion and a subsequent penalty before the Warriors cut the gap to two points after 11 minutes.
A driving tackle by Scott Barrow on Saracens full-back Brent Russell set up the position for Parks to get the hosts on the board with a penalty.
As the half wore on Glasgow did their best to play with pace and width and they added two more Parks penalties, but Jackson replied in kind for the visitors and he slid the knife home in first-half injury time.
Andy Farrell was a typically robust presence in the Saracens midfield
The fly-half read an inside pass from Glasgow scrum-half Sam Pinder and strolled over for an interception try from halfway, with his conversion making it 18-9 at the break.
Glasgow's hopes were now hanging by a thread and Saracens twice came close to finishing them off in the third quarter.
First Haughton was held up over the line, Parks redeeming himself for some poor passing by wrapping him up in a fine tackle, and Russell then dropped the ball a yard from the line after Jackson and Andy Farrell had made the initial incision.
Time was running out for the hosts but on the hour mark they finally escaped their own 22 and promptly scored a sparkling try.
After winning quick ball off the top at a line-out Thom Evans stepped past Haughton and the young winger, fresh from being called up to Scotland's Six Nations squad, sent Stortoni over.
Jackson took the gap back out to seven points with his third penalty to settle Saracens nerves as Glasgow gave it everything in their battle to keep their campaign alive.
John Beattie spoiled a fine break by not giving the try-scoring pass to the supporting Pinder before Parks added a fourth penalty to cut the gap to four points once more, but the Londoners held on to create their own little bit of history.
In the other Pool Four match Biarritz defeated Italian side Viadana 25-16 to clinch second place in the group.
Glasgow captain Alastair Kellock:
"The biggest frustration for us is that we handed Saracens their two tries - and that was the difference.
"In terms of effort and application, I could not have asked for any more from my guys, but we let ourselves down when it came to making mistakes and executing certain decisions.
"Considering what we have put into the campaign physically and mentally, it is hugely disappointing to be eliminated like this."
Saracens captain Neil de Kock:
"The result was a huge weight off our shoulders. The quarter-finals were always our aim for the season, and now we can think about going even further because we have a home tie.
"If we had been offered that at the start of the campaign, we would certainly have taken it.
"We knew Glasgow would make it very hard for us as they had done every time we played them in recent times."
Glasgow: Stortoni; Evans, Gregor, Barrow, O'Hare; Parks, Pinder; Va'a, Thomson, Low, Palepoi, Kellock, Brown, Barclay, Beattie.
Substitutes: Hall, Tkachuk, Kalman, Turner, Eddie, Morrison.
Saracens: Russell; Haughton, Sorrell, Farrell, Penney; Jackson, De Kock; Lloyd, Ongaro, Johnston, Jack, Vyvyan, Chesney, Hill, Skirving.
Substitutes: Kyriacou, Mercey, Ryder, Gustard, Dickens, Powell, Ratovou.