The Whites went into the Elland Road clash on the back of consecutive Sky Bet Championship defeats for the first time since September.
But goals from Chris Wood - his 22nd of a prolific season - and Pablo Hernandez ensured they got their play-off push back on track.
"It was a good response and pleasing to get the three points," Monk told LUTV.
"There was a little bit more pressure and expectancy on us to get the result tonight.
"It was the first time we had suffered back-to-back defeats since the start of the season, so we are happy to get the win.
"I thought we did very well in the first half in particular and we are pleased to get back on track"
Fifth-placed Leeds dominated the first half but, having doubled their lead in the 47th minute, allowed City back into the game and were reliant on a fine performance from Rob Green.
The former England goalkeeper denied David Cotterill, Joe Bryan and substitute Milan Djuric, clawing the latter's effort off his line with one hand.
Green was eventually beaten at the death, Djuric heading in a corner in stoppage-time, but it proved to be the last action of the game.
"Bristol created a few chances and Rob Green made some good saves for us," Monk added.
"Five extra minutes were issued, their goal came in the sixth, so it was disappointing as the game should have been over."
Bristol City boss Lee Johnson was unhappy that Wood's goal was allowed to stand after the striker appeared to commit a foul on Bailey Wright in the build-up.
"The first goal is always going to be big, especially at a big away club like this," Johnson said on the club's official Twitter account.
"The first goal is a clear foul. Wood has clipped Bailey Wright's heels and pushed him in the back."
City, now only three points above the relegation zone, suffered a blow when 21-goal striker Tammy Abraham limped off towards the end of the first half with a thigh problem.
Johnson believes the injury is a bad one for the on-loan Chelsea striker, saying: "Tammy Abraham has done something to his thigh. He'll have a scan over the next day or two.
"Tammy is gutted because he feels like it might be a bad one but hopefully that won't be the case."