Saracens, European champions in 2016 and 2017, were the only team to win all six pool matches this season, including two against Glasgow, but the first try went to the visitors in less than two minutes at Allianz Park.
Scotland fullback Stuart Hogg straightened the attack before feeding wing Rory Hughes, who found scrumhalf Ali Price to finish a stunning move.
The hosts responded through fullback Liam Williams to draw level inside five minutes, before Alex Lozowski, who took over kicking duties from Farrell, knocked over a penalty to give them the lead.
Momentum swung firmly in Saracens' favour as David Strettle and Brad Barritt forced their way through the Glasgow defence for two tries in four minutes.
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Two penalties from Glasgow flyhalf Adam Hastings reduced the hosts' lead to nine, but second-half tries for Williams, Strettle, Jamie George and Nick Tompkins sealed a convincing win for Saracens.
Saracens, who won the European title in 2016 and 2017, will play Ireland's Munster in the semi-finals in Coventry on April 20.
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
