Ross Wallace's deflected shot, Chris Iwelumo's 100th career goal and a penalty from Graham Alexander, who was making his 900th club career start, gave the Clarets victory at Turf Moor.
Laws said: "I thought we were a bit sloppy in that first period and had Leicester had their shooting boots on they could have gone two up.
"There is still room for improvement and there are areas which I'm disappointed with. However, the important thing is getting a foothold on the game, getting the first goal is key, the manner in which you get it doesn't matter.
"Last season we conceded some horrendous goals from deflections and today it was one of those that got us going.
"It gives a confidence to the players and we stepped up after that and were more penetrative with the ball, passed it better and moved it quicker."
Burnley sit third in the Championship after their opening three games and although there is an air of confidence at Turf Moor, Laws was quick to point out that there is plenty of work still to do.
He added: "I'm delighted with the start. Two clean sheets in three is something really solid to build from.
"It will give the players confidence, even more so the supporters and the work we have put in during pre-season is paying dividends at the moment, but there's a hell of a long way to go yet."
Saturday's win comes on the back of an opening-day victory over much-fancied Nottingham Forest, and a draw at Ipswich last time out, and Laws admits the signs are good.
"We said Leicester, Forest and Ipswich were always going to be contenders and it would give us a yardstick as to what this division is about," said the Clarets boss. "One thing we have to be is very organised, mistakes will be punished.
"On a better day Leicester would have got something out of it.
"Over the 90 minutes we justifiably won the game and hopefully we'll take that into our next game.
"We're moving in the right direction, that's for sure."
Leicester boss Paulo Sousa, who is still searching for his first win in charge of the Foxes, said: "We did start well and had two great chances to score in the first 20 minutes.
"When you play a team like Burnley these opportunities make the difference."
Sousa's message to the fans was simple - keep the faith.
"They need to keep belief," he added. "If they reflect a bit, they will realise we are close to a win. I hopes things turn for us - soon."