The Blues returned to Cardiff Arms Park searching for their first competitive victory of the season following a pair of defeats.
Danny Wilson’s men made a flying start to the encounter and they led by 10-points early on thanks to a fine effort from Tom James and the boot of Steven Shingler.
Shingler and Finn Russell exchanged shots at goal before the Scottish outfit struck against the run of play.
The Blues fly-half added a further penalty after the break but again the Warriors struck through Lee Jones.
That was to be the telling score as Shingler and Russell exchanged further penalties before Shingler fired wide at the death.
Glasgow dominated the opening five minutes and following a penalty kick to the corner they set up camp on the Blues’ five-metre line.
The Scots went through a succession of phases but they were met by some ferocious home defence.
Cardiff Blues met Glasgow with impressive line-speed and physicality forcing Finn Russell to opt for the cross-kick.
Alarm bells rang as Glasgow had superior numbers on the left but Cuthbert back pedalled and soared into the skies to claim a superb mark.
Minutes later the home side struck with virtually their first attack as Tom James dummied to the outside and then fended Alex Dunbar before racing 40 metres for his first try of the season.
Shingler converted the solo score and added a difficult penalty on 12 minutes following a fine scrummaging effort from Rhys Gill and Keiron Assiratti.
The penalty gave the Blues a 10-point lead after 12 minutes.
Glasgow hit-back five minutes later with a Finn Russell kick at goal and Dave Rennie’s side continued to threaten.
They were awarded a fortuitous penalty soon after, which Russell fired into the corner but the lineout was overthrown and Josh Navidi rampaged out of his 22.
Cuthbert displayed his electric pace mid-way through the half as he raced after a long Shingler kick.
Lee Jones managed to dot down in the nick of time but he felt the full force of Cuthbert in the process.
Shingler kept the scoreboard ticking but Glasgow responded with a fine try of their own on 32-minutes, with some simple interplay down the short side putting Henry Pyrgos away.
Russell converted to cut Cardiff Blues’ lead to just three points but it could have been much worse.
Glasgow broke through from inside their own half, thanks to Dunbar, but the Blues clung on and Seb Davies won a potentially try-saving turnover.
The Blues went into the break with their slender three-point lead but Shingler quickly doubled the advantage after the restart.
However the Blues were dealt a major blow just two minutes later as Russell beat Rhun Williams to a high ball to create a try finished by Lee Jones.
The Scotland fly-half converted the effort to give his side the lead for the first occasion but rather than drop their heads the Blues came back fighting.
They quickly threatened with Rey Lee-Lo slipping a couple of tackles and finding Lloyd Williams in support.
Williams was unable to find a way around Glasgow’s covering defence but the Scots blatantly infringed at the breakdown, allowing Shingler to reclaim the lead.
Russell booted the Warriors back in front on 67 minutes, a just reward following an extended period of pressure.
The Blues had opportunities of their own, one in particular off the back of a scrum where Lee-Lo spun out of contact and popped a pass.
However, the ball slipped out of Shingler’s hands as he looked to make the pass with an overlap on the outside.
Cardiff Blues continued to threaten as they chased the encounter, but a late missed penalty by Shingler, who had maintained a 100 per cent record up to that point, saw the visitors run out 19-20 winners.