Two goals in five early second-half minutes had put the Clarets in control of the Championship clash, with Dean Marney and Chris Eagles finding the net.
Sheffield United rallied, though, and hit back through Daniel Bogdanovic's first goal for the club and Ched Evans' deflected effort.
Burnley looked to have sealed the win when Jay Rodriguez's 90th-minute strike restored their lead, only for Mark Yeates to cap a man-of-the-match performance with a sweet strike from the edge of the area and earn a 3-3 draw for the Blades.
Losing the two points was a bitter pill for Laws to swallow on his return to Sheffield, where he managed across the city at Wednesday.
And with Burnley still without an away win this campaign, he knows his side cannot afford a repeat if they are to achieve their promotion target.
"Any manager would say if you're 2-0 up away from home you expect to go on and win the game," Laws said.
"I don't think anyone could have seen how the game was going to pan out.
"If we want to do well in this division we have to come away from home and finish games. When you're 2-0 up you want to win games but I thought we lacked a bit of organisation and leadership.
"We defended too deep, it felt like the defenders were playing behind the goalkeeper at one point.
"I'm very disappointed and the players know that. Sheffield United work hard and if you give them a sniff they will take it.
"We know what we want to achieve this year and this is a big lesson to take on board and I don't want to see it repeated."
Sheffield United manager Gary Speed was pleased to see his side take a point, although he was unhappy that they found themselves in need of the late rescue act after pulling back to 2-2.
Former first-team coach Speed has endured a tough start to life as a number one, with his side unable to muster a win in any of their last four games.
And he knows a winter of struggle could be on the cards if they keep finding themselves in such precarious positions in games.
"I'm feeling a mixture of frustration and anger and pride," Speed said.
"Losing two goals after half-time wasn't good, but to get back to 2-2 saw great resolve and commitment which was great.
"We then let another one in which was criminal and there's no excuse for that, but great attitude again to get a third.
"But you can't keep climbing mountains and slide down and expect to climb up again. But if we show the commitment we have then the luck will turn to us."