Glasgow head coach Sean Lineen was confident his players would bounce back against Llanelli in the Celtic League.
The Warriors slumped to a shock defeat to Connacht last week but Wales stars Stephen Jones and Mark Jones got the Scarlets off to a flying start but the Warriors rallied through the boot of Scotland stand-off Dan Parks and tries from Thom Evans and John Barclay to go in one point ahead at the break.
The home side then stretched their cushion with a try from Test hooker Dougie Hall before another try Scotland hopeful Thom Evans put the Firhill outfit in sight of victory.
Mark Jones added another try in the 72nd minute but replacement scrum-half Colin Gregor sealed the victory for Glasgow.
Lineen said: "I'm so pleased for the guys. We had a good sit down after the Connacht game and they put in a lot of hard work before this game.
"Even when Llanelli came back into the game, I knew we had it in us to win the game. The feeling in the dressing room before the game was electric, I could just feel it."