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.
Then 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 secured a home quarter-final.
Early on 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. But 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.
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.
On the hour mark the hosts 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.
Tries: Stortoni. Pens: Parks 4.
Tries: Haughton, Jackson. Cons: Jackson. Pens: Jackson 3.
Stortoni, T. Evans, Gregor, Barrow, O’Hare, Parks, Pinder, Va’a, Thomson, Low, Palepoi, Kellock, Brown, Barclay, Beattie.
Morrison for Gregor (48), Tckachuk for Va’a (48), Kalman for Low (71), Turner for Palepoi (41), Eddie for Brown (74).
Russell, Haughton, Sorrell, Farrell, Penney, Jackson, de Kock, Lloyd, Ongaro, Johnson, Jack, Vyvyan, Chesney, Hill, Skirving.
Ratuvou for Sorrell (24), Gustard for Hill (51). Not Used: Kyriacou, Mercey, Ryder, Dickens, Powell.
Alan Lewis (Ireland).