Jamie Roberts, Rhys Williams and Tal Selley crossed in an impressive first 40 minutes from the visitors.
Tries from Tal Selley, Martyn Thomas, Rhodri McAtee and Alec Jenkins saw them to a comfortable 26-5 victory.
Yet within eight minutes the Scarlets were back down to 14 men, after Selley was judged to have deliberately knocked-on.
Selley claimed his second try of the match in injury time, Bowen converting, while Glasgow had a consolation score from replacement Lawson.
Selley grabbed his second try after the break and Mark Lewis added the fifth late on, both converted by the excellent Nick Robinson.
The City boss was also critical of Dorchester's approach to the game, which saw former Arsenal midfielder Ian Selley sent off for two bookable offences.
The second period was all Wales though, and tries from Alec Jenkins and Lee Beach set up a thrilling climax that saw Tal Selley's try steal victory.
Summer signing Richie Rees made the break and centre Selley, on loan from the Ospreys, provided great support and had enough pace to beat the covering defenders.
Aled Thomas cut the gap to 3-13 with his first penalty, but the Dragons defence was cut to pieces minutes later as Selley opened his try account for the season.
In the group stages Wales - who won the bottom-tier Shield competition in Dubai last week - secured an excellent 14-14 draw with England, James Harris and Tal Selley scoring late tries.
Robinson hit the post from the resulting penalty kick at goal, but the ball fell kindly to the Blues and they eventually created an overlap for Selley to dive over in the right corner.
Most of those Welsh tries came in the win over Chinese Taipei, with Scarlets duo Alec Jenkins and Jon Edwards crossing twice, and Tal Selley, Rhodri McAtee, Dai Flannagan, Richard Mustoe and Lee Beach also touching down.
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