The home side were 2-0 up at the break in the Premier League encounter after two goals from crosses, the first an own goal by goalkeeper Brian Jensen before Christian Benitez netted his first goal at St Andrew's.
Steven Thompson pulled one back late on but it could not save the Clarets from another defeat.
Laws was disappointed with the award of the free-kick that led to Birmingham's second goal, but conceded his side were their own worst enemy.
He told Sky Sports: "The disappointing thing is, with the ball we were the best side, we passed the ball very well. But we haven't defended two crosses.
"When you're down at the bottom end you never get the rub of the green. For the second goal it was a fantastic tackle from Steven Caldwell."
Burnley already knew they would be playing Championship football next season after their relegation was confirmed with defeat by Liverpool last weekend.
Laws added: "We know what we've got to do. There's a lot to work on, we've got to work on both boxes but there's plenty of good play here as well."
For Birmingham, the result ensured they maintained their unbeaten home run, which stretches back to September.
Midfielder Barry Ferguson said: "It was important to stay unbeaten at home. The boys have been excellent so it's great to end (at home) with a win. The first half we were by far the better team but the second half petered out a little bit."
The Blues were tipped to struggle in their first season back in the top flight but they have defied all expectations and today's win means they are likely to finish ninth.
"At the start of the season we would have taken fourth bottom," said Ferguson. "There's a great team spirit here so to finish in the top 10 would be a great achievement."
Looking ahead to next season, the Scotsman added: "Hopefully the manager will get a bit of money to spend on a few quality players and hopefully we can go on from here."