Saracens showed it is going to require something very special to wrestle the Champions Cup out of their grasp as they ran in four tries to make it 16 games without defeat in the tournament.
You have to go back to the semi-final in 2015 for their last defeat, 13-9 by ASM Clermont Auvergne, and they are into the last four for the fifth successive season. Now they will meet Munster Rugby at the Aviva Stadium in the semi-final.
Glasgow arrived with a small army of around 6,000 fans to cheer them on in their first quarter-final, but they were never able to gain enough possession, or create enough phases, to break down the Saracens defence.
They managed to stay in the hunt for an hour, Lee Jones' second half try providing them with their best moment as they cut the gap momentarily to six points in the second half. It would have been even closer had Finn Russell not hit the upright with his conversion attempt.
But Saracens kept their composure, dominated the final 20 minutes and ended with a flourish of three tries from centres Marcello Bosh and Brad Barritt, and flying wing Chris Ashton. Ashton scored in both halves to take his European Cup tally to 36 to draw level with Vincent Clerc as the leading scorer.
He could also have got a third had the TMO not ruled out his acrobatic effort to score in only the third minute. Fellow wing Sean Maitland also had a try ruled out in the first half.
There was no stopping Ashton when he sniffed the line a second time, rolling over in the tackle when he was hauled down just short to score. Owen Farrell, named by Stuart Barnes as the Heineken Man of the Match, added the conversion and his boot helped to make it 14-3 at the break.
That Jones try gave Glasgow a glimmer of hope, but Saracens were in no mood to give another inch after being caught out by a deft cross kick from Russell to Jones across the 22 after a scrum. They picked up the pace and stormed back into the game to take complete control with tries from Bosch, who beat four defenders as he weaved his way over from the edge of the 22, and Barritt.
Ashton then equalled Clerc's record with a typical corner finish and Farrell added the extras to take his match tally to 18 points. Ryan Wilson grabbed a consolation try for Glasgow in the last move of the match.
