A fast start saw Shingler and Hadleigh Parkes each cross over, while tries for Mike Cusack and Glenn Bryce sought to get Glasgow back into the contest.
Scarlets went on to finish the match with just 11 players on the pitch but they held firm - making more than 100 second-half tackles - to open their account for the season with a valuable victory over the GUINNESS PRO12 champions.
"We were really pleased, especially after last year's form away from home," Shingler told the club website. "It was good to get four points given the way it finished. The boys on the field held on exceptionally well.
"We'll take a massive amount from it, defensively we're going to be stronger for what we went through in that second half and it's exciting times.
"The defence was huge. In the first two games last season, we lost through fatigue in defence, slipping off tackles. It's still early days but we really have to back it up this weekend."
This weekend throws up a home clash with Ulster, who also began the GUINNESS PRO12 season in impressive fashion, beating the Ospreys 28-6.
Shingler added: "Every time I've played Ulster, they've been a strong team.
"They have a strong squad and it will be tough for us on Saturday. If we can come through that with another four points, it will be pleasing."
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