Man City looked far more likely to equalize for Ferran Torres for the home side, but Dallas’ breakaway goal in added time gave Leeds the win.
It was the fourth league defeat of the season for Pep Guardiolas Man City, who are 14 points ahead of a crucial Champions League game against Borussia Dortmund next week.
Leeds only had two shots during their entire Saturday game compared to City’s 29, but it was enough to take the win and get into the top half of the table.
“It’s a great result for us,” Dallas told BT Sport. “It was tough out there today. It’s tough enough against any team of 11 men, regardless of Man City at 10.
“It was difficult, we had to get used to it and I thought we defended very, very well.”
Dallas gave Leeds the lead in the first half when his low shot rolled off the post from the edge of the box.
But when Cooper’s challenge against Gabriel Jesus was improved from a yellow to a red card shortly before half-time after a review by a video assistant, Leeds were faced with a long 45 minutes against the outstanding team in the Premier League this season.
Torres eventually leveled when he found his place in the box and converted Bernardo Silva’s pass, and City continued to put pressure on Leeds’ goal for the final 15 minutes of the game.
But the visiting side, who have played a number of high-scoring games since their promotion to the Premier League, stubbornly defended.
Raphinha was late on goal only to be superbly attacked by City goalkeeper Ederson.
A moment later, however, Dallas caught the winner from yet another break in Leeds when he shot his shot through Ederson’s legs.
It was Pep Guardiola’s first loss to a team led by Marcelo Bielsa – two coaches who share a close relationship and value each other.
After a 2-1 win in the first leg, City is now traveling to Dortmund to reach the semi-finals of the Champions League.
With four trophies this season, it only takes 11 points to claim the Premier League title.
Leeds, meanwhile, face a difficult series of games with Liverpool, Manchester United and Tottenham Hotspur three of their next four opponents.
But after beating Man City at Etihad, Bielsa’s side will feel like they can beat anyone.
Red ones leave it late
Elsewhere in the Premier League, Liverpool ended six straight Premier League defeats at Anfield with a 2-1 comeback win over Aston Villa.
Liverpool, who suffered a humiliating 7-2 loss to Villa last October, were left behind after a goal from Ollie Watkins during the break, while Roberto Firmino’s goal at halftime was also ruled out by an offside under construction.
Mohamed Salah equalized as he headed Andy Robertson’s saved shot and the game seemed to end in a draw until Trent Alexander-Arnold’s great stoppage time gave the Reds a much-needed win at Anfield.
Salah’s goal was the first Liverpool had scored at home in the league since February 7, but Jurgen Klopp’s men seem to be leaving that lousy form behind with three consecutive Premier League wins.
The result lifts Liverpool, who will try to break a 3-1 deficit against Real Madrid in the Champions League quarter-finals next week, to fourth place, while Villa is in 10th place, one place below Leeds.