A 40-9 victory away to Zebre Parma last weekend, combined with Leinster's defeat at the Stormers, lifted Franco Smith's men four points clear at the top of the URC table with three rounds of the regular season to go.
Josh McKay - who returned from illness to start at full-back against Zebre - says there's plenty of belief in the Glasgow squad as they prepare to face Bulls and Lions in South Africa.
"Absolutely. There's a lot of belief in this group," McKay told Scotland Rugby News. "Everyone is motivated. It would be a massive step forward for the club. But we've still got a big job ahead of us in South Africa and back at home when we play Zebre again [on the final day].
"There are still three big games left before we start putting our eyes on that. The excitement is definitely there in the group to go and do the job."
The Bulls are currently fourth, seven points behind Glasgow, and the Lions in touch of a play-off place, so the double-header will be a huge test for Warriors, but McKay urged his team-mates to seize the opportunity.
"We've put ourselves in a good spot to go deep into this competition. You definitely can't take that for granted," said the full-back.
"We're all riding it until the end, everyone is going all in on it. We're no longer in Europe.
"I think the way the competition is going to finish up, leading into play-offs will suit us with a couple of weeks off where we can really hone in on the training but also get fresh with no games at the weekend.
"South Africa is going to be huge for us and can be a good springboard into the tail end of the season."
The top four earn home quarter-finals, the top two potential home semi-finals, and the regular season winners a home final if they make it that far.
Warriors haven't lost a home URC game this season. Their only defeat at home was against Northampton Saints in the European Champions Cup in December.
"Of course, any team is going to want to play at home over going away aren't they? We would love to play as many games at home as possible in front of our friends and family," McKay said.
"But we don't want to look at it from far back, we just want to take it one game at a time and by doing that, to keep it simple, we've just got to keep winning, not get too far ahead of ourselves."
The depth in the Glasgow squad has been a big factor in Glasgow's progress this season with young players like Gregor Hiddleston, Max Williamson, Euan Ferrie and Jamie Dobie stepping up.
"You need a big squad that is ready to go and can turn up on any day, because it's such a long season with both competitions, Europe and the URC," McKay added.
"It takes a toll on a lot of boys. We lose a lot during the Test windows as well.
"I think Franco has done a really good job at blooding a lot of players. A lot of players are not just getting one or two games a season, there are a lot getting good-quality minutes and when they step in, they're doing a good job.
"The depth is crucial... you need a good squad to go deep into these competitions."