The Whites took a first half lead through Raphinha's penalty, the first of three spot-kicks awarded over the course of the afternoon, before a poor giveaway in their own half allowed Chelsea to hit back through Mason Mount.
Leeds fell behind on 58 minutes when Jorginho scored from the spot following Raphinha's rash tackle on Antonio Rudiger, but fought back to level seven minutes from time when Tyler Robert's cross was tucked in by substitute Joe Gelhardt.
The teenager's first Leeds goal appeared to have earned a point for the visitors until Mateusz Klich fouled Rudiger in the area and Jorginho beat Meslier again from 12 yards.
"We would have deserved a draw, we had merits to deserve it," said Bielsa after the game.
"We were able to create danger, it was nice to watch. For us to establish differences was a great challenge. We had to defend but we did it well and it was difficult for us to attack but moderately we created chances.
"No game is alike but the performance today makes me think we can recover the version that allows us to compete with the best. "
Bielsa refused to lay the blame for defeat at the feet of Raphinha and Klich for giving away penalties.
"There are a high percentage of actions that are unexpected and errors are possible in the game," he said.
"Unless they're linked to being uninterested or they are unjustified or impudent, from my optic I wouldn't classify those two penalty decisions like that."
Leeds were already without Patrick Bamford, Liam Cooper, Kalvin Phillips and Robin Koch before Pascal Struijk picked up a knock 24 hours before the game, focing Bielsa into a late tactical change.
"[Pascal] was going to play the game, in yesterday's training he had a movement, two of the bones in his foot collided and that produced pain," said Bielsa.
"It has no risk but, pain is difficult to tolerate. The inflammation didn't go down from yesterday to today and last minute we couldn't count on him.
"[Shackleton] was in the position in which Stuart Dallas was going to play, the defence was going to be Luke Ayling, Diego Llorente, Pascal and Junior Firpo. When Pascal wasn't able to be under consideration, Dallas played in place of Ayling, Ayling and Llorente played as a pair instead of Pascal and Llorente and the role in front of Ayling, behind Raphinha was given to Shackleton. He is a player we imagined could reproduce what Dallas was going to go in that position."
The head coach's decision to send on Gelhardt was almost instantly rewarded, the striker scoring with his first touch much to Bielsa's delight.
"It was a goal that reflects his qualities," said the Argentine.
"Being in the right place at the right moment, to confirm those characteristics at this level is important, especially as it was a goal that allowed us to level the game. It was not the fourth goal of a thumping, it was 2-2 away from home against Chelsea."
Players from both sides were involved in a heated confrontation after the full-time whistle, Firpo being led away as Rudiger ran across the field to get involved with the left-back and then Llorente. Bielsa saw little in the way of serious infractions, though.
"In a game with emotions that were so high, you have liberating passions," he said.
"We don't hope for these things to happen. It was within the margin of what was tolerable. All of us will have hoped it didn't happen. We play in grave limits."