Saturday's win came on the back of an alarming start of the season for the Tykes, who were thrashed 4-0 at QPR in their opening Championship game then beaten 1-0 by League One Rochdale in the Carling Cup.
But Patrick McCarthy's own goal gave the Reds a much-needed victory.
Robins said: "I thought we were decent, especially after the week we've had. I love watching them in training, I think their movement is good and when they settle down and really start to believe they'll be a really good side. It's important to get a win because you've got all the outside pressures which are heaped upon you.
"We're trying to make a big, big change from what was here before and I think you could see signs of it, but it's not what they're capable of producing.
"We kept Palace very quiet and restricted them to nothing.
"The only disappointing thing was that we had to rely on an own goal.
"I would have loved one of our boots to have put the finishing touch to that move. But it will give the lads confidence to carry on and really have a go."
The decisive goal came in the 35th minute after Hugo Colace won the ball in midfield and Diego Arismendi picked out a wonderful first-time ball to Jay McEveley, whose driven cross was diverted beyond stranded Julian Speroni by McCarthy.
Robins singled out on-loan Manchester City midfielder Kieran Trippier for special praise.
He said: "Keiran is an outstanding footballer and that was his position, he started out as a right winger. But he can play anywhere because he's a really good footballer.
"He's filled out from when we had him last year and I'm delighted to have him. I think he's going to be a top player."
Barnsley: Steele, Hassell, Foster, Shackell, McEveley, Trippier (Devaney 77), Colace (Butterfield 79), Arismendi (Doyle 68), Hammill, Lovre, Gray. Unused substitutes: Preece, Dickinson, Morales Neumann, Potter.
Crystal Palace: Speroni, David Wright, Davis, McCarthy, Clyne, Ambrose, Cadogan (Danns 46), Garvan, Dorman (Djilali 81), Lee, Zaha. Unused substitutes: Price, Barrett, Andrew, Bennett, Pinney.
Referee: Neil Swarbrick (Lancashire).
Man of the match: Keiran Trippier.