Bristol City boss Lee Johnson labelled Leeds "lucky" after the game, citing a controversial red card for Josh Brownhill.
An under-strength Leeds opened the scoring on 69 minutes when Kemar Roofe turned home a Pablo Hernandez strike from close range. It was game over when Hernandez headed home Samuel Saiz's perfect cross four minutes from time, meaning Leeds took the points on a day when Bielsa had to patch a side together.
He lost his two senior goalkeepers, Jamal Blackman and Bailey Peacock-Farrell, ahead of the game, resulting in a debut for academy stopper Will Huffer. He was lacking several defenders too, meaning the untried Aapo Halme was thrown in at the deep end.
But the game dramatically swung in United's favour when Brownhill was red-carded for two bookable offences on 55 minutes. Oncethe Robins were down to 10 me, Leeds stepped up their game.
Roofe drilled an effort wide on 59 minutes after getting past a defender and into the box and Gjanni Alioski could not find a finish from a tight angle on the hour-mark after a good pass from Mateusz Klich.
Klich drilled an effort straight at the keeper seconds later before Leeds finally got the breakthrough. Bielsa, who also has a number of attacking players missing as he deals with an injury crisis, insisted his side were worthy.
"It's a point of view that we have to respect, he said. "But we virtually didn't concede any chances for our opponent.
"As far as I know I don't remember anything of note that they created. It was a very comfortable debut for our goalkeeper today. When I see the game again I will take this into account this point of view that we didn't deserve the result.
"That we played with one player more made it easy for us. But that wasn't the only reason we won. We did not dominate the end of the first half or the start of the second half but we were dominant from that point on.
"I am pleased with the performance and the win today. I thought were were very good when we had the man advantage and really created plenty of chances."
Bielsa's counterpart believed his side were unlucky not to take anything from the game.
"I thought they (Leeds) were lucky today," said Robins' boss Johnson. "I thought they got away with it. We were the side who looked like going on to win the game until the red card. The referee took the easy decision in front of a big crowd. The referee wants to run the show these days.
"There were decisions that we didn't get but I don't care about the refs because they do it to us every week. We have had some poor refs the last few weeks.
"There are plenty of positives we can take from this. We were right in it until the red card and were in a good position to get a result. We were the better side once we settled. We were breaking well and our shape was good.
"We did not get enough shots on target, we are disappointed but we feel we got our game back and we have to encourage the players."