While Grant believes that win was among his proudest achievements for Edinburgh, the 28-year-old admits his side still have a lot of ground to make up on a Warriors side currently topping the GUINNESS PRO12.
Yet with Edinburgh claiming impressive back-to-back victories over Benetton Treviso and Scarlets in recent weeks, Grant is confident his team are moving in the right direction.
"We're sick of hearing about Glasgow, but as much as guys argue or complain, they've been the better performing team than us for years now, which makes it more frustrating for us," Grant told BBC Scotland.
"The 1872 Cup was one of the best moments of my Edinburgh career. We've wanted that so long and worked so hard for it.
"Do we think we're not as good as them? Not a chance! But figures and stats show they are the better team.
"That's our drive - to be the number one performing team in Scotland."
While Glasgow have flourished under the leadership of head coach Gregor Townsend in recent years, Grant insists Edinburgh are building a similarly tight unit under boss Alan Solomons.
"This is the tightest I've felt Edinburgh in all my years," he said. "We've had a lot of overseas guys, but it's the culture, we've got key guys who have been here a while.
"It's come from the top as well, Solly (Alan Solomons) has been a huge part of creating this environment.
"When you get guys in, it doesn't matter where they're from, they fit into that environment, and if they don't, they're out.
"We've got a brand now, defensively we've got an identity, we're very physical."
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