Wingers George North and Andy Fenby were on the try score sheet again - both now with three tries a piece from two RaboDirect games; also on the score board was athletic back rower Aaron Shingler who was in the thick of the action at the Friday night fixture.
Easterby said he was proud of this players at a top-billing game where the home crowd were desperate for a win for Glasgow to mark a new era in their new stadium.
But it was not to be for the Scottish side who reached the semi-finals of the competition last season as with mental and physical toughness, the Scarlets set about their task with a level-headed and determined performance- executing a well-thought out game plan and finishing their opportunities clinically.
A late surge by Glasgow saw them come back into the game with a try but the cushion created by the Scarlets with a lead of 18-6 in the second half allowed them to retain their composure and strangle-hold on the game.
Once again, the Scarlets put previous statistics aside and set the record straight with a win in Glasgow - something that has escaped them in recent years with just one win from seven of their last contests with the Warriors. The Warriors lost just one of their last nine matches against the Welsh regions last season.
An all-important away win early in the competition at a challenging fixture will give the Scarlets another positive boost and great heart as they go into their second away game in Connacht next weekend - with just two weeks until the big derby contest at Parc y Scarlets over rivals the Ospreys on Friday 21st September (kick off 7.05pm).
Simon Easterby said he was pleased with his side's overall performance: "I am very proud of the way the players secured that win, they worked very hard for that victory, it was hard-fought and showed the character and commitment in this Scarlets team.
"You can't ask more from the players than to come up to Glasgow and secure a tough win especially on a big night and occasion for the Warriors and their home supporters.
"We finished off our opportunities well, our attack looked lively and to get three tries away from home was particularly encouraging; we also defended well for large chunks of the game and the boys showed solid determination and mental strength because physiologically it was always going to be a difficult game for us.
"We kept our hold on the game and executed the game plan well; we knew they'd throw everything at us on Friday night and we held firm.
"We soaked up their pressure in the second half well I thought the boys came out of the changing room at half time with a real edge; mentally and physically we were a match for them and it takes a certain mindset to do that away from home so there's positives that we can take from that game now in our preparations for our trip to Connacht next weekend.
"We've got off to a decent start in the league but it's just that, a start - and there are also some important areas of our game for us to work on going forward."
NEXT HOME GAME - THE BIG DERBY IN THE WEST
SCARLETS V OSPREYS, Parc y Scarlets, Friday 21st September kick off 7.05pm. Be there and support your Scarlets at one of the big Welsh rugby events of the season.