With Sheffield United having won their early kick-off against Rotherham, Bielsa’s men needed all three points at Ashton Gate to retain second place behind leaders Norwich.
It is looking increasingly like a three-way battle to go up automatically. But Bielsa declined to admit as much, despite being more than satisfied with the way his side saw off play-off chasers City in a rousing, but error-strewn match.
He said: “It is a risk talking about even being safe in the top three at this stage. There are so many tough matches ahead.
“Today ended up a 50-50 game. In the first hour we could have scored more, but in the final 15 minutes our opponents could have got a draw.
“At the end they played lot of aerial balls into our box where the consequences of mistakes can be high.
“My team played with a lot of experience in that period when we couldn’t launch counter attacks.
“Our goalkeeper Kiko Casilla brings serenity to the team because he does not make mistakes whatever situation he faces.
“Kalvin Phillips did well in a role protecting the back-four. We had worked hard on our aerial play down the middle of the pitch.”
The decisive goal came in the ninth minute, Bamford sliding to score from a header across goal by former City defender Luke Ayling and hurting himself when colliding with a post.
“I could have taken him off earlier than I did, but he is a very important player to us,” said Bielsa, who eventually substituted the striker early in the second half.
Leeds were the better side for long periods. It took the hosts until first half stoppage time to create a chance and then Famara Diedhiou fatally delayed his shot from Marlon Pack’s pass.
Tyler Roberts should have made it 2-0 to Leeds in the 57th minute, but shot straight at City goalkeeper Max O’Leary from 10 yards.
At the other end Casillas blocked Pack’s shot with an outstretched leg and substitute Matty Taylor failed to lob the keeper when he was caught off his line.
But the home side did not create enough to deserve anything and have beaten Leeds only once in 14 Championship meetings since 2010.
Head coach Lee Johnson said: “I was disappointed with our quality in the first half, but you have give them some credit for that.
“It was a horrible scrappy game and Leeds did to us what we have done to a number of opponents this season.
“Our substitutes had an effect with good cameo appearances, but we have to be better collectively from the start of games.
“It was a poor goal to concede and we will look at that. But in the end Famara Diedhiou and Matty Taylor have missed good chances to get us a point.
“Now we face Ipswich here on Tuesday night and that won’t be easy because they have taken a good point at West Brom today.
“Our destiny is still in our own hands as far as the play-offs are concerned, which is important.
“The fans will be vital to us during the Ipswich game because we will be expected to win and that won’t be easy. No Championship game is.”