Speaking after the game at the Millennium Stadium on Saturday, Williams said it was so important that the region got a good home support in their remaining trio of games in the RaboDirect PRO12.
He said: "The boys have worked hard to keep us in the fight for a play-off place; it's exciting times for us now in these last three games and we are all really chuffed to still be in the mix for a semi-final.
"The edge that our supporters can give us will be vital in these last few games - it's important that we make the most of our home advantage and a run-in to the final that sees us stay at Parc y Scarlets throughout."
With home ties against table toppers Glasgow, Cardiff Blues and then finally Treviso, the Scarlets who are tied with the Ospreys on an equal 58 points, have everything still to play for.
Williams said: "We really appreciate the commitment our supporters give us, and even in some challenging periods we are very lucky as a region to have such loyal and passionate fans."
Scott Williams said he had enjoyed returning back to the region after the triumph of the Six Nations.
"We were really chuffed with the result on Saturday - we'll have a big emphasis on trying to improve our accuracy as there were a few moments when we were playing with such confidence and trying things - and some of the stuff we were doing didn't quite come off.
"If we keep hold of the ball, we showed how dangerous this side can be when it's firing and it was good that we finally took the game away at the end on Saturday.
"We dominated for the first 30 minutes and our possession and territory was around 65% so we just needed to convert our pressure into points - which happened in the end.
"We have a good run in now, and we wanted to say a massive thanks to all the supporters for what they have done in coming to the Millennium Stadium and backing our region.
"The key is now that we are still in the fight for the top four and for the players returning from Wales, it's great that we have a real focus and momentum for the end of our season.
"We have back to back home games to finish - none of them are easy but as a home team we have to make it tough for teams to come to Parc y Scarlets and our home support can be part of that."
Williams said Judgement Day had been a success for Welsh regional rugby.
He added: "It was a fantastic occasion to be a part of on Saturday for everyone involved - it was great for all four regions to play out in front of a massive crowd and thanks to all the regional rugby supporters for being part of it."
Scott Williams said it was an excellent opportunity and experience to give some of the region's younger players including Scarlets half-backs Owen Williams and Aled Davies to play in a senior regional game at the national stadium.
"There were plenty of Millennium Stadium debuts made in regional rugby on Saturday and that can only be good for our game for wider players to experience big match day events like that."
Guinness PRO12 Scarlets News
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