Yet they mounted a late rally against the Eagles which harvested two goals in the last 10 minutes from midfielder Sean McAllister and substitute Leon Clarke.
A relieved Laws said: "It wasn't about our performance today - it was about getting the win and avoiding a third straight defeat. That was the key.
"Even before the game, our back four was put together with some glue, sellotape and sticky back plastic.
"Richard Wood had to pull out before the game because he wasn't fit and we were very, very stretched.
"I had to put Frank Simek in the side and he's only played 70 minutes in a year.
"That's credit to him, and Richard Hinds has had to play a second game when he should never have been anywhere near the side.
"But I thought they were magnificent."
Laws paid tribute to his goalscorers after the game exploded into life in the 80th minute when McAllister took matters into his own hands.
The 21-year-old, one of the Owls' better performers, kept his composure to chest the ball down 12 yards from goal and sweep a fierce left-foot shot into the bottom corner.
McAllister has been marginalised in recent weeks but he justified his recall.
Laws said: "He's been disappointed not to be in the side but this was probably the perfect time for him to play.
"I needed to freshen things up and we know what Sean is all about.
"He's got great energy and covers every blade of grass and he can pop up in little areas and get a goal."
Clarke added a second in the third minute of stoppage-time with a close-range tap-in to give the scoreline a disjointed look.
Laws added: "You open Leon's tool box and he's got it all.
"Sometimes you just don't see enough of it, and I felt it was justified that he was on the bench today because he hasn't played as well as would like to over the last couple of games.
"However, that will give him a massive lift; coming on and making an impact, and I thought we stretched the game even more as the game went on.
"Yes, we did have fortune on our side as they hit the post twice, but I'm just delighted to get the win, especially for the supporters as they have backed us 100 percent.
"Wins breed confidence and that will breath confidence into the side."
The result was harsh on Palace, who hit the post in either half through Victor Moses and Clint Hill.
Warnock pinpointed the loss of centre-back Claude Davis to injury in the 56th minute as the turning point in the game.
The former Sheffield United boss said: "I thought it was there to win and we've ended up losing it, but I thought as soon as Claude Davis went off we got back to old ways.
"I thought that was the difference. When he was on we were comfortable but when he came off we were a good 20 yards deeper and invited more pressure.
"But if Victor Moses or Clint Hill had scored and not hit the post I think we would have been comfortable.
"The first goal was always going to win it. I thought we should have stepped up a gear and tried to win, but that's something we're obviously not capable of."