Lucy MacRae's early penalty put Glasgow ahead but an opportunist try by Sioned Harries, converted by Molly Wilkinson, put the visitors in front
Glasgow enjoyed spells of possession and territory but couldn't turn that into points and Thunder's Chloe Thomas-Bradley went over to give the visitors a 12-3 lead at the break.
Claudia McLaren hit back with a fine individual effort but tries from Shona Wakely and Meg Webb put Thunder further ahead.
Glasgow never gave up and a fine kick pass from Ceitidh Ainsworth found Roma Fraser who went over for the try.
MacRae again added the tricky conversion, but the visitors kept Glasgow out for the remaining 15 minutes to seal victory.
The match was part of the second phase of the Celtic Challenge, the play-off stages after five rounds.
The six teams have now split, with the top three sides playing for first, second and third and the bottom three fourth, fifth and sixth positions.
Glasgow and Brython have joined the other Welsh side, Gwalia Lightning, in the bottom half.