Farrell was withdrawn from the starting line-up to be with his wife after she went into labour, and it proved to be a double celebration for the Sarries talisman, with his side's seven-try rout setting up a last-four showdown against Munster.
His absence meant a reshuffle as Alex Goode filled in at five-eighth, with Liam Williams at full-back, and that duo combined to score the try that wiped out Glasgow's opener.
Ali Price had already capped a fine Warriors move with the game's opening score before Williams collected Goode's smart high kick to bundle over the line.
An Alex Lozowski penalty gave Sarries a lead that was extended to 22-13 by the interval, David Strettle — another beneficiary of Farrell's pre-match departure — and Brad Barritt helping themselves to tries, while Adam Hastings replied with six points from the boot.
The hosts gained a measure of comfort early in the second half from Lozowski's penalty and Williams' second try of the afternoon.
It was game over when Stafford McDowall's loose pass was intercepted by Strettle to double his personal tally before the hour mark, with Jamie George also joining the party.
A George Horne try briefly stemmed the tide before Nick Tompkins added further gloss with a score between the posts, although Glasgow did at least have the final say when Matt Fagerson crossed.
Up next for Mark McCall's men is Munster, which overcame Edinburgh 17-13 with a dogged display in the Scottish capital.
The visitors were immense in defence, although Chris Dean did touch down for Edinburgh as Keith Earls's opener was cancelled out, with Jaco van der Walt's penalty handing Richard Cockerill's men the half-time advantage.
But Munster stuck at its task and, after Tyler Bleyendaal and Van der Walt had exchanged penalties, Earls put the finishing touches on a fine team move before Bleyendaal nervelessly converted.