They picked up their fourth win in six league games on Sunday, running in four tries to beat struggling Connacht 32-30 at Parc y Scarlets.
Three of their final four RaboDirect PRO12 games are also at home and, after opening up a seven-point gap back to Connacht in seventh, Easterby is confident of staying put.
"This win was massive for us," he told WalesOnline.
"The great thing about this competition is that if top six is what we have got to get to, it makes these games all the more interesting.
"We have missed out on the top four because of our early-season form and losing games we should have won, but the most important thing is that we finish strongly with three home games and the Blues at the Millennium Stadium to come.
"We have got to get as many points as we can and finish as high as we can and at the moment, qualification for Europe is in our own hands."
And the Scarlets boss was able to cool suggestions that scrum-half Gareth Davies is facing a lengthy spell on the treatment table.
He was forced off after 55 minutes of Sunday's win but Easterby said: "He has a dead leg, which hopefully won't keep him out for too long.
"It was a nasty one and the initial reaction was to make sure we get him off. Yes, the early consideration was that it might have been a break, but in his words, it was 'a mental dead leg'."
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