Laws made nine changes from the side which beat Leicester at the weekend but saw his second string come back from conceding a shock early goal to see off a Morecambe side who dumped Coventry out of the competition in the first round.
Phil Jevons stroked home his first goal of the season to give the League Two side the lead before Chris Eagles fired home Steve Thompson's lay-off to equalise.
Thompson made no mistake from the spot and Kevin McDonald's long range shot found the net to give Laws a welcome win.
He said: "We made a number of changes but I was confident that with the strength of our squad we could do that.
"Morecambe started well and took the lead which gave us an early test but we showed real spirit and I think we proved what a good squad we have.
"It was always going to be a tough game and it had every potential to be an upset. But to be fair to the players they dug in and although it wasn't pretty at times we did enough to win the game.
"Once we scored the equaliser we played the football I know we are capable of and passed it well and it was a good win."
Morecambe boss Sammy McIlroy was left to bemoan his side's luck after losing their early lead.
He said: "We had a great start and took the lead and could have had a second not long after which would have made things really interesting.
"But Burnley scored a good goal to come back into it and it was always going to be tough after that although we did continue to cause them problems.
"We hit the crossbar and had a couple of half chances but we gave away a soft penalty again and that gave Burnley the advantage they needed.
"I was pleased with the way we battled right to the end but when they scored a third with a cruel deflection it was game over."