Luke Ayling scored the only goal of the game early on and despite dominating the remainder of the match and missing a host of chances, Leeds held on to stay second in the Sky Bet Championship.
With other promotion rivals faltering, Bielsa will hope his side can now put together a run that sees them end their 16-year exile from the top flight.
He certainly believes they are more experienced than 12 months ago, when they eventually squandered second spot and ended up dropping into the play-offs, bowing out at the semi-final stage.
"We are now experienced in every type of situation in the Championship," said Bielsa. "I feel we have more resources to manage the different kinds of matches.
"But I cannot forget this season we were winning 3-0 and drew the match (against Cardiff).
"This was a fair win. We attacked well, we missed a lot of chances. But we defended well and they created few chances.
"The performance, in general for the team, was positive."
Leeds made a bright start and after a few near-misses, they opened the scoring after 15 minutes.
The impressive Ayling, up against his former club, stabbed the ball home inside the box after a mad scramble that saw three efforts blocked before it fell kindly to the full-back.
Leeds thought they had doubled their lead just a couple of minutes later when Patrick Bamford tapped in after a shot had bounced kindly for him after striking the crossbar.
Unfortunately, the flag was raised immediately as Bamford was offside.
Undeterred, Bamford then fired a shot at goal after a superb control and spin.
City started brightly after the restart and were appealing for a penalty when Ayling blocked a shot in the area. Referee Tim Robinson was not interested despite vocal appeals from City boss Lee Johnson and his coaching staff.
That was as lively as it got for City, though, with Kiko Casilla hardly threatened in the Leeds goal.
The Whites bossed the second half, but their profligacy was again on show.
A minute later Bentley made a scarcely-believable save. Costa latched onto a long pass and rounded Bentley only for the City goalkeeper to somehow stick out a hand and divert it behind for a corner.
The impressive Jack Harrison thought he was the one to finally grab a second goal but his speculative effort came back off the crossbar.
Bristol City boss Lee Johnson said: "I thought that Leeds were the better side throughout. In the first 15/20 minutes we didn't accept that we needed to do certain things and have a bravery to go at the game.
"At half-time I wasn't best pleased. That's not discrediting Leeds because they were very good. We just didn't play like I want us to play.
"We did, for most parts of the second half, and I thought we made a game of it. We've just got to be better against the top teams."