In a hard-fought encounter, Warriors went down 31-25 but score five tries to take two points from their visit to Edinburgh.
Glasgow started strongly and opened the scoring through Beth Blacklock, before extending their lead when Hannah Smyth crossed in the corner. Edinburgh responded through Aila Ronald, with Hannah Ramsay converting, but the Warriors struck again via Sky Phimister.
Late first-half pressure from the hosts paid off as Molly Poolman's try, converted by Ramsay, left Edinburgh trailing by just one point at the break (14-15).
Edinburgh took control after half-time. Tries from Hannah Walker, Cieron Bell and Nicole Marlow, with Ramsay again accurate from the tee, stretched the lead to 31-15.
Glasgow rallied in the closing stages, with co-captain Holland Bogan scoring twice to set up a tense finish, but Edinburgh held on to secure the victory.
At full-time, player of the match Bogan praised her side's effort, saying: "It was a really difficult match. Edinburgh were always going to be tough at home, but I'm proud of the team and know we'll keep building throughout the competition."