Aaron Shingler, Andy Fenby and George North all crossed for tries as the Scarlets spoilt the party at the Glasgow's new Scotstoun home by building on their 45-point hammering of Heineken Cup holders Leinster in Round 1.
"We said we needed to back up last week's win over Leinster by going away and getting a result and the team showed that mentally, as well as physically, they were up for the game," said Easterby.
"I'm very proud of the way the players secured that win, they worked very hard for the victory.
"They executed the game plan well and to score three tries away in Scotland - which has been a difficult place for us in the past - shows the character and intent of this team this season. We finished off our opportunities well and our attack looked lively.
"The boys showed solid determination and mental strength because psychologically it was always going to be a difficult game for us. We kept our hold on the game - we knew they'd throw everything at us on Friday night and we held firm.
"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."
It's been a superb opening to the season for a team that have just missed out on the play-offs two years running but Easterby is refusing to get carried away.
The former Scarlets flanker, who was part of the backroom staff before taking over from Nigel Davies this summer, knows there's plenty of work to do to ensure they secure at least a semi-final spot this time around.
"We looked comfortable as a team, our defence was good for large chunks of the game and overall it's a strong and positive start to our season," added Easterby.
"There's plenty to work on, though, and we have a tough challenge ahead of us out in Connacht next week, followed by the Ospreys at home on September 21 for what will be a high pressure derby fixture at Parc y Scarlets - but these are the types of big games our players want to compete in.
"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."