Rhodri Williams, Scott Williams and Jordan Williams all crossed for the Scarlets in South West London as the region produced some breath taking rugby to jump to the top of Pool 4.
But while he was clearly delighted with both the performance and the result in Round 1, Easterby admits his team still have a huge battle to make it through to the quarter finals given that French giants Racing Metro and Clermont Auvergne are also in the group.
"It's one win out of six pool games. We will not be getting too carried away. It's a fantastic start, but that is all it is at the moment," said Easterby.
"We've got a pretty tough group. We've got Racing up next and it's vital you win your home games. If you get points on the road then that gives you a big advantage and we've done that.
"But this won't mean anything if we don't go home and perform and produce the type of display that we did against Harlequins and get a result next weekend.
"We have to keep our feet on the ground and realise we are only one game into the competition. We will have to come back next week and produce a similar type of display."
The Scarlets' Heineken Cup hopes had been widely written off even before a ball had been kicked this season, with the big-spending Top 14 duo and triple Amlin Challenge Cup winners Quins expected to ensure they finished bottom of their pool for second season running.
Wales centre Jonathan Davies has suggested those external criticisms could spur the region on prior to Round 1 and Easterby says he never doubted his players' ability to step up on the big stage.
"No-one gave us a chance apart from ourselves and we've enjoyed that this week a little bit, coming away first up and we've shown the sort of rugby we can produce and backed that up with some intelligent play and good defence at the end," added Easterby.
"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've spoken about lifting ourselves for this competition because we've been there before as a club and a region and we wanted to experience that again.
"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 here."