Wins away to Clermont Auvergne on Saturday and at home to Saracens the following weekend represent a formidable challenge - but one that is far from impossible. Should Glasgow rise to the occasion, a path featuring home fixtures all the way to the semi-finals could begin to take shape.
"I suppose the ball is in our court now," Glasgow assistant coach Nigel Carolan told the media on Tuesday.
"Taking five points at Sale and another five against Toulouse has really shifted the outlook on the competition.
"The URC matters, but we've done that the hard way. If you finish third or fourth, you're usually on the road - possibly even for a semi-final. We've been there before; it's no longer unfamiliar territory.
"But the chance of securing a home round of 16 in the Champions Cup, maybe even a home quarter-final, means there's a lot at stake. We won't look beyond this weekend, though. We rested some key players with this game in mind, and now we're going all guns blazing."
Head coach Franco Smith has consistently tried to "keep expectation out of our environment" this season as he rebuilds the squad following the summer departures of Tom Jordan, Sebastian Cancelliere and Henco Venter.
The dramatic comeback against Toulouse - overturning a 21-0 deficit to defeat the six-time European champions - has inevitably fuelled hopes that Glasgow can compete with Europe's elite when it matters most.
"At the start of the season, especially after losing players you consider front-liners, the big question was whether we had the depth to compete on two fronts," Carolan said.
"But that all changed after the Toulouse win. It showed that maybe we can.
"It's crucial that when we go to Clermont, we don't try to protect what we've got or retreat into our shells. Ten points is nothing to defend at this stage.
"If you want a home round of 16 or a home quarter-final, you have to put your best foot forward from the opening whistle. We need to be brave in how we play and make sure we're not holding anything back."
Since a shock 23-0 URC defeat by Scarlets in November, Glasgow have built real momentum, reeling off five straight wins. That run includes a derby double over Edinburgh Rugby and those two emphatic statements on the European stage.
There is a palpable feel-good factor around the club, which could be boosted further this weekend by the return of two influential players from injury.
"We're fortunate that Huw Jones and Kyle Rowe are both back training, so they're very much in the mix," Carolan said.
"They've worked incredibly hard on their rehab and are very close to returning.
"They just need to tick off one or two contact elements. They've been on the field already, just not doing everything yet.
"It's touch and go whether one or both might be involved later in the week—but we're holding our breath and hoping we might get both."