Two tries from Chris Ashton and one apiece by Marcelo Bosch and Brad Barritt kept the champions on course for a successful title defence, man of the match Owen Farrell contributing four penalties and three conversions.
Ashton advanced and damaged his Lions’ prospects in the space of 18 minutes, producing a magnificent finish only to then expose his fragile defence by waving over wing Lee Jones.
Saracens led 14-3 at the interval and apart from a spell early in the second-half when Glasgow threatened a fightback after crossing through Jones, the outcome was never in doubt.
Munster are Sarries’ next European assignment and in three weeks’ time they must visit the Aviva Stadium, the setting for Wasps’ collapse before Leinster on Saturday.
Glasgow were forced to make two try-saving tackles at the corner flags inside the opening five minutes after Saracens plundered an early turnover and counter- attacked with precision. First Jones kept out Ashton and then Stuart Hogg denied Sean Maitland, although on the second attempt the champions might have been better served had Alex Goode looked inside to Richard Wigglesworth.
But Glasgow settled and began to make an impression on the home defence for the first time only to trail to three Farrell penalties. Not for the first time Saracens were stripped of the ball, Wigglesworth being dispossessed by Henry Pyrgos to bring one promising attack to a close, but on the half hour mark the visiting whitewash was breached.
Waves of runners made dents before Billy Vunipola made a bulldozing run that created a chance that still needed plenty of finishing. Barritt sucked in two defenders and slipped a superb ball out of the tackle to Ashton, who drove forwards before rolling over the line.
The race to the whitewash was on when space was created for a second Ashton try only for Hogg to nudge the wing into touch.
Open side Jackson Wray had an eventful start to the second-half, preventing Finn Russell from capitalising on an opportunist hack downfield and then flattening Jones under the high ball with a challenge on the cusp of illegality.
Saracens are famed for their ‘wolfpack’ defence, but it was exposed in the 49th minutes as Jones took revenge for the hit from Wray by scoring a soft try.
Russell lofted a kick into the left corner and Jones pounced, catching the ball and then evading feeble grabs from Goode and Ashton to touch down.
The stage was set for a Glasgow comeback, but instead they were breached for a second time 10 minutes later. A clearance reached only as far as Ashton and following runs from Vunipola and Barritt, the ball was fed to Bosch who stepped his way past the white shirts and over.
Barritt touched down as the beneficiary of a superb offload from Schalk Brits and Ashton grabbed his second before flanker Ryan Wilson scored a consolation in an action- packed final 10 minutes.
ries: Ashton 2, Bosch, Barritt. Cons: Farrell 3. Pens: Farrell 4.
Goode; Ashton, Bosch, Barritt, Maitland, Farrell, Wigglesworth, M. Vunipola, George, Figallo, Itoje, Hamilton, Rhodes, Wray, B. Vunipola.
Lozowski for Goode (79), D. Taylor for Bosch (60), de Kock for Wigglesworth (75), Lamositele for M. Vunipola (72), Brits for George (55), Koch for Figallo (28), Conlon for Hamilton (72), Brown for B. Vunipola (75). Glasgow scorers: Tries: L. Jones, Wilson. Pens: Russell.
Hogg, Seymour, Dunbar, P. Horne, L. Jones, Russell, Pyrgos, Reid, Brown, Fagerson, Alainu-uese, Gray, Harley, Wilson, Ashe, Peterson.
Grigg for P. Horne (57), Price for Pyrgos (51), Allan for Reid (60), Flynn for Brown (60), Puafisi for Fagerson (48), Peterson for Gray (11), Fusaro for Peterson (71). Not Used: Hughes.
Jerome Garces (France).