The home side came back from 19-15 down to beat the Tigers with late tries from wing Andy Fenby, full-back Gareth Owen and centre Nic Reynolds.
"A home win has been a long time coming," Easterby said. "We have come in for a bit of criticism and some of that has been justified. We have rewarded our fans with a win and hopefully we have set a template for the next three or four weeks."
He added: "Coming in to this game there was a bit of pressure on us because we were coming up against a pretty experienced Leicester team. We took our opportunities in the first half in terms of some excellent kicking (from Owen Williams) and kept the scoreboard ticking over.
"They scored a couple of tries from our error and indiscipline but the important thing was we kept in the game. Owen Williams kept putting three, six, nine points on the board. That was important because we set the platform for the second half."
Leicester coach Richard Cockerill admitted he was disappointed with his side's discipline in giving Scarlets 21 points in penalties, while prop Fraser Balmain and lock Rob Andrew were sin-binned in the second half.
"We weren't good enough and we weren't disciplined enough and the Scarlets took advantage of that and credit to them for that," Cockerill said. "It was a dead rubber and we are not used to those sort of games but we still wanted to win it. It was disappointing. The result looks a bit one-sided.
"We gave 21 points away in penalties. We scored three tries in the first half but we were only four points up because they kicked their goals.
"The Scarlets played really well fair play to them. We got two sin-bins in the second half and that hurt us. The momentum went their way and they punished us.
"I was disappointed in our discipline and there were one or two guys who consistently gave penalties away, and that is a lesson for us."
- Sports & Recreation