Sheffield, who won Friday’s first leg 83-76 to lead the tie by seven points, went down 72-96 at Ponds Forge on Sunday afternoon.
That meant Glasgow progressed to next week's semi finals thanks to an overall scoreline of 172-155 in their favour, which means Sheffield's six-year wait for a title continues.
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Sheffield had to make do without influential guard Rodney Glasgow Jr due to a hamstring injury picked up in the first leg.
Point guard Antwain Johnson was also missing following Achilles surgery before the season finished.
The home side looked off colour from the start.
Visitors Glasgow, with nothing to lose, set out to reduce the deficit immediately from tip-off and opened up a 10-point lead in a first quarter which ended 29-24 in their favour.
The scoreline only widened in the second period with Sheffield trailing by 16 points at one stage before back-to-back three-pointers from Kipper Nichols brought them back to within 10.
Sharks coach Atiba Lyons called two time-outs in the second period, but was unable to change the trajectory of the match as Glasgow continued in the ascendancy and went into the half-time interval ahead by 54 points to 42.
Sheffield rallied after the interval, however, and won the third quarter 23-16 to lead the tie by two points overall heading into the last 10 minutes.
But Glasgow dominated the final period and produced 26 points to the Sharks’ seven to see out the match comfortably.