The west Wales region had failed to pick up a single point in losing their first four BKT URC fixtures, leaving them bottom of the table.
In contrast, Glasgow had won four of their opening five games to stand in third place.
But, despite being depleted by having five backs away on Wales duty, the Scarlets tore up the form book to claim their first victory of the season.
There were tries from flanker Josh Macleod and centre Macs Page, plus some real heroics in defence to keep Glasgow scoreless in the Llanelli rain.
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Delighted head coach Peel said: “It feels great. We were quite honest in our review of the first block. We had a lot of work to do and we have worked hard over the last couple of weeks.
“I'm happy for everyone because we obviously haven't had the best start to the season.
“I thought we did the basics well. We kicked well, we chased well, we turned ball over and we took points when we needed to.
“Scoreboard pressure is huge in those conditions. You don’t want to be points down chasing the game because it’s difficult to move the ball.
“I am pleased for the boys because they have given a lot of hard work. I am really happy with the performance and the win. It’s a good start to a really exciting block.”
Skipper Macleod said: “I asked for a level of intensity. We knew it was going to be won on defence and I think nil on the scoreboard is a pretty decent success.
“We knew as forwards we had to step up and we felt a bit of dominance up front.
“Hopefully this is the building block - nil on the scoreboard and some decent tries.
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Overall we are just very, very happy to get the win. It’s a pretty decent foundation moving into an exciting block.”
There was also a win for Cardiff who beat Zebre 29-14 out in northern Italy to take them to the giddy heights of second in the table.
Hooker Daf Hughes scored a first half hat-trick of line-out drive tries and then replacement flanker Taine Basham stepped his way over to seal the bonus point in the final play of the game.
Head coach Corniel van Zyl said: “Zebre are a tough team to beat, so we will definitely take it. I am happy for the boys to get it over the line.
“We had to dig deep and the boys did it, so I am very proud of them.”
Hat-trick hero Hughes, who was named Player of the Match, added his thoughts on a win achieved without 10 Wales squad members.
“It’s a tough place to come. Your pack really needs to be on fire to take a win and we were. To come out here and get five points is massive,” he said.
“We found out last year that there’s one or two points in it to be in the top eight. If we keep our heads down and go game by game, we can put ourselves in a good spot come the end of the season.”
As for his hat-trick, he said: “I can’t take too much credit. The boys in front of me did well and I was at the back end of the maul.”
Despite being without 23 players, the Ospreys put in a heroic display as they picked up a losing bonus point in a narrow 19-17 defeat away to Edinburgh, while the depleted Dragons also pushed champions Leinster hard before going down 24-10 at Rodney Parade.
