Three opportunistic tries and a perfect kicking record for outside-half Duncan Weir were enough to put Glasgow comfortably ahead before an Alex Dunbar break for Glasgow got the fourth try to secure a comfortable win.
Glasgow opened the scoring with a try by scrum-half Colin Gregor. It was made by full-back Stuart Hogg, who emerged from a maul when several would-be tacklers seemed to let him go.
Hogg was suddenly left in the open to give the scoring pass to the supporting Gregor.
The Blues hit back with a Ceri Sweeney penalty before Sweeney turned creator, a clever cross-kick going to centre Gavin Evans who slipped the ball back inside for fellow centre Casey Laulala to score.
Glasgow got their noses back in front when flanker James Eddie ran 40 yards for a try from the base of the scrum, number eight Johnnie Beattie taking the ball wide and then passing back inside to the flanker.
Glasgow ended the half with a series of scrums on the Blues line. Eventually home scrum-half Lewis Jones was given a yellow card for trying to get at the ball from an offside position as the home pack were under pressure.
Glasgow outside-half Duncan Weir kicked the penalty to go with his two successful conversions and maintain his perfect kicking record as the visitors went into half-time with a 17-8 lead.
Sweeney had a chance to narrow the gap straight after the break, but his penalty hit the outside of the post from 35 yards out.
That was the end of the match for Jones. When his 10 minutes were up, the Blues brought on replacement scrum-half Richie Rees, with centre Dafydd Hewitt coming on for Evans in a double substitution to try to change things.
However it was Glasgow who went further ahead with right wing Tommy Seymour intercepting a laboured pass by Blues lock Deiniol Jones and sprinting 40 metres to the line. Weir converted once again.
The Blues stepped up their pressure and that paid off when Sweeney charged down a defensive kick from Hogg, flanker Michael Paterson leaping highest to take the rebound and flop over the line to score the try. Sweeney again missed the conversion and soon gave way to Gareth Davies.
Glasgow scrum pressure brought them another penalty and Weir was able to add his fifth kick from in front of the posts to put the visitors further ahead.
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