Bamford scored the only goal of the game the eighth minute at Ashton Gate but injured himself in the process.
Regardless, Bamford played on until the 57th minute when Bielsa eventually substituted him.
And the Argentine has explained that he kept the forward on because of the impact he has on games.
“He is such an important player to us,” Bielsa said.
“Bamford’s (injury) was an impact injury and it prevented him from reaching his maximum speed. It’s not a muscular injury.
“I could have taken the decision (to substitute him) earlier but for us we need to make movements on the pitch to attack and press the opponent.
“Bamford is very important in this and because he’s so important I didn’t want to take him out of the game.
“But I saw his performance weakened us a little after the goal.”
LEEDS PLAYER RATINGS vs BRISTOL CITY – TWO STARS GET 5/10
Bielsa does not expect Bamford to be out for long as Leeds look to hunt down Norwich in the Championship.
The Elland Road outfit are two points behind the Canaries after 36 matches.
All sides are in action in midweek with Leeds making the trip to Reading on Tuesday (8pm). Norwich will host Hull the following night (7.45pm).
Midfielder Kalvin Phillips twisted his ankle during the win over Bristol City, though played the full 90 minutes.
And Bielsa is hopeful both he and Bamford will be available for the trip to the Madejski.
Bielsa said: “We’ll see by Tuesday how they are.
“I think they will be available but it wouldn’t be sensible to predict their situations.”