Rhys Patchell had crossed in the first half and kicked a penalty, while another Wales international, Leigh Halfpenny, contributed 16 points with the boot.
The shock victory over reigning champions Toulon in October had encouraged Blues supporters to dream of success in Europe despite mediocre results in the RaboDirect Pro12. This success will have raised their hopes even further.
Despite the return of Halfpenny and Cuthbert from Wales duty, the Blues were without six others - including national captain Sam Warburton - injured while representing their country in the four-match autumn programme.
The Warriors had had their Scotland men back for two weeks, but made nine changes from the side which beat the Ospreys last weekend.
Glasgow showed the early positivity, but it was the Blues who came close to the first try when flanker Macaulay Cook broke and sent Cuthbert down the touchline, but opposite number DTH van der Merwe saved the day
Halfpenny was narrowly wide with a penalty, but made amends when another Glasgow offence allowed him a second chance from virtually the same spot, before both teams came near to tries.
Warriors prop Ryan Grant knocked on under the Blues posts and, after a Cuthbert kick and chase, Alex Dunbar knocked the ball out of the winger's hands as he went to touch down.
Young fly-half Patchell doubled the Blues lead with a monster penalty from well over halfway, before Glasgow fly-half Dougie Weir knocked on behind his own line and Patchell darted over from the resulting five-metre scrum.
Halfpenny converted and added another penalty as the Welsh region went 16 points in front.
But Glasgow, desperate to cut the deficit before the interval, piled on the pressure, with mercurial Fijian scrum-half Niko Matawalu a constant threat, and they were rewarded when Lions wing Sean Maitland popped up at outside centre to step inside two tacklers and touch down. Weir added the extras with the last kick of the half.
A sniping run by Blues scrum-half Lloyd Williams ran out of support and Patchell was short with an optimistic drop-goal attempt, but the home side still collected the first points of the second period thanks to a long-range Halfpenny penalty, only for Weir to reply with a similar effort.
Warriors skipper Alastair Kellock was held up short by Cuthbert, apparently knocking on, but referee George Clancy gave the visitors the scrum and it needed a crunching tackle from centre Dafydd Hewitt to deny Weir. But the TMO ruled it was high and Weir picked himself up to slot over the penalty.
Another Halfpenny penalty extended the Blues lead to two scores with just 10 minutes left and Cuthbert put matters beyond doubt when he intercepted a loose pass deep in his own half and raced 75 yards to score, handing off replacement Ruaridh Jackson to get there. Halfpenny converted and it was effectively all over.
Glasgow had the last word with a try by Fijian replacement Leone Nakarawa, improved by Weir, but it was no more than a consolation.
- Sports & Recreation
- Leigh Halfpenny
- Rhys Patchell