Only the heroics of City goalkeeper David Bentley, some last ditch defending and the woodwork twice prevented the Whites from winning by five or six, with the statistics telling their own story as Leeds had 21 shots to three and 12 corners to nil as well as recording their usual high possession count at just short of 70 per cent.
Apart from maybe some of the finishing, there were few signs of the nervousness getting to the Whites on this occasion as they dominated their visitors in every department, carrying on the good all-round work of their midweek game against another of the contenders, Brentford.
And with results elsewhere going Leeds' way they now have a three-point cushion once again in second place with only one of the top seven still to play this season.
Head coach Bielsa was pleased with the effort his players put into the game.
He said: "We attacked well, we missed a lot of chances. We defended well, they created few chances. We fought for every ball.
"The performance in general and collectively was very positive.
"The team controlled the match, defended very well, attacked very well, we scored just one goal, you saw all the chances the team had.
"We ran a lot. We recovered a lot of 50-50 balls."
United were rarely threatened in the game, apart from one chance for Nahki Wells when he hit the side netting in the 80th minute, but when asked if Leeds had learned to manage games better, Bielsa was cautionary.
He explained: "We have experienced every type of situation in the Championship and I feel that we have more resources to manage the different kind of matches we face.
"But I can not forget in this season we were winning 3-0 (against Cardiff) and drew the match.
"Every player recovered the ball a lot of times. Technically the team showed a good level as well. We didn't lose important balls. We imposed as a team in the one v ones, the dribbles, the balls in behind.
"From the first minute we felt the support of our fans and that was throughout the match.
"I agree that the team showed character on the pitch and if there was tension the team didn't realise that."
Leeds were on their game from the first minute as they tore into their visitors to win a first minute corner from which Helder Costa had a shot deflected wide.
From the resulting second corner Liam Cooper had a good chance to follow up his midweek goal against Brentford, but headed wide.
Within a minute Costa - who went on to have his most effective game in a Leeds shirt - raced in behind the City defence and dribbled to the bye-line only for his low cross to be cut out by keeper Bentley with Patrick Bamford behind him and ready for a tap-in.
It was no surprise when the Whites opened the scoring in the 16th minute after an almighty scramble. Four efforts went wide or were blocked in an incredible attack before the ball fell to full-back Ayling, who took a touch in the area before burying a left foot shot.
It could have been two soon after when Costa cut in from the right and saw his shot beaten away by Bentley. The follow-up effort by Stuart Dallas was deflected onto the crossbar and the rebound put in by Bamford only for the striker to be correctly ruled offside.
It was not Bamford's day in front of goal, but he produced an impressive display to be a constant thorn in the visitors' defence and earn praise from Bielsa. He was unlucky again when he came up with a fine turn on the edge of the area only to fire wide.
Mateusz Klich and Costa brought further saves from Bentley with strikes from just outside the area while Bristol finally managed their first effort as Jamie Paterson's tame header was easily collected by Kiko Casilla.
The second half saw the visitors look more determined to impose their game on the hosts as they began to run with the ball with more purpose to begin with. Shouts for a penalty for handball against Ayling fell on deaf ears and a long range strike went harmlessly wide.
But United regained their dominance and could not believe they could not quite put the game to bed in a red hot spell midway through the half.
Bamford should have done better with a clear chance after meeting Costa's pull back with a shot that was half-blocked by Bentley then cleared off the line.
Jack Harrison's cross was just too strong for an unmarked Costa at the far post then Costa showed fantastic pace to get on the end of a Cooper long ball, taking the ball round Bentley with his first touch before the keeper somehow recovered to block the winger's shot.
Dallas sent a 25-yard shot over before City had their one genuine opening as Wells was slipped in behind the home defence, but from a tight angle could only shoot into the side netting.
It was business as usual with Leeds on the attack soon after and it was Harrison's turn to curse his luck as he smacked a powerful shot against the crossbar with Bentley well beaten this time.
New striker Jean-Kevin Augustin looked a little lost after coming on for Bamford for the last 15 minutes, but he was presented with one chance in added time after being found in the area by Kalvin Phillips and showed quick feet to get a shot away that was only inches wide.
The 1-0 final score was remarkable in the circumstances, but the performance was the key with Leeds looking back to the team we saw before Christmas. Now the trick is to repeat this quality with another home game next Saturday against Reading.
Leeds United 1
Bristol City 0
Leeds: Casilla; White, Phillips, Cooper; Ayling, Klich, Dallas; Costa, Hernandez (Shackleton 90), Harrison; Bamford (Augustin 75).
Bristol: Bentley; Kalas, Baker, Williams, Dasilva; Eliasson (Palmer 70), Massengo, Henriksen, Paterson (Diedhiou 60), Weimann (O'Dowda 34), Wells.
Referee: Tom Robinson.