On a night when Leeds were not at their best, they dug deep and were rewarded in stoppage time in dramatic fashion when Palace defender Marc Guehi was penalised for handball after raising an arm after jumping with Liam Cooper following a corner.
Raphinha tucked away the penalty in the third minute of stoppage time to score the only goal of the game as United moved up to sixth from bottom - five points away from the side occupying the final place in the relegation zone in Norwich City, who drew at Newcastle United.
Bielsa, whose side secured just their second home victory of the campaign, said: "It was a victory that was very necessary. In every game there are the points at play that can improve our campaign.
"The team fought for the whole game and that is true and necessary. Either way our team usually gives everything in every game. In that aspect, it was very similar to the game versus Leicester and a victory always brings happiness and in a human group, happiness is indispensable.
"It was a relief and we can breathe again and it is very satisfying. But the succession of games means that we have another challenge once again.
"The intensity in which the team played and the constant search for a goal, even though it was difficult to get it and the closeness and togetherness of the team to defend and attack, shone through.
“After the goal there were still five minutes to play. Only at the end did we feel the win was ours.
“We haven’t taken many points so far this season. Every game we see as an opportunity to balance this out.
“The opposing team had developed their play very well and it was very difficult to keep that clean sheet.
“To have won the previous game always helps to prepare for the next one. Sometimes a win facilitates the next one.”
On Raphinha holding his nerve, he continued: "He’s very important and to score that goal, he needed a lot of character, personality and courage. He got us the three points."
Leeds attitude was summed up by the spirit of the likes of Tyler Roberts and Dan James, whose industry and heart on a difficult night shone through as Bielsa's side chiselled out a precious win.
On Roberts, handed a first start since mid-October, added: "It was important. He did not ever stop fighting. He kept our team alive and present and I didn't imagine he could run as he did. It was a performance that allowed him to score he was an option.
"I don't think it was the changes that changed the win was going. (Mateusz) Klich had made a big effort, similar to Tyler's and Rodrigo was another option in the same position and a bit more fresh. We ended up defending well and contributing down the sides."