The holders of the Celtic League trophy will visit Hughenden on Friday 3 October in the quarter-final of the new knock-out competition.
"We had been hoping for a home draw, and that is what we've got," David Jordan, Glasgow's chief executive, said.
"Of course, it will be demanding match against the current league champions, but we're already looking forward to it."
Glasgow won their way through to the last eight with an away win against Celtic Warriors by 19-9 last night. "Our victory at Bridgend proved we have their ability to rise to the challenge, whatever the opposition," the chief executive added.
Glasgow winger Dave Millard helped Aberdeen GSFP to a 35-20 win over Hawick at Mansfield Park. Dave scored a try to give Aberdeen a bonus point to take them to the top of BT Premiership 1.
Glasgow's Calvin Howarth, Cammy Mather and Lee Harrison were named in the BBC Wales Scrum V Team of the Week for the Celtic Cup weekend.