The Scarlets lost all six games in last year's pool stages and have been up and down in their domestic league so far this term but they were simply sensational in shocking the former English Champions on their own turf.
"The Heineken Cup is special and it brings out the best in you. The feeling around the competition is you have to go up to another level. We talked about that before the game," said Easterby.
"We weren't perfect, but we were close to producing some of the best rugby we've played this year. We didn't do ourselves justice last year. But we did ourselves justice today.
"We have to enjoy this win because it was a big performance. It's great for the players and fans. At times, you would have thought it was a home game for ourselves because of the way that the fans reacted and supported the boys and gave us a lift throughout the game."
The Williams' trio of Rhodri, Scott and Jordan all crossed to hand the Scarlets a flying start to Pool 4 but Easterby was keen to praise the collective effort rather than single out individual stars.
"I couldn't pick out one player because they all gave their all. The players deserve a huge amount of credit.
"We knew Quins would come back and give us the game we expected, but, credit to the lads, we showed character at the end.
"We had 14 men twice in the game, but we scored six points when we had a man in the bin in the first insistence and then we didn't concede the second time. We dug in and we deserved the result."