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Phil Foden gave City the lead in the seventh minute when he fired home from Rodri’s lay-off, before Erling Haaland headed in a second in the 15th minute from Matheus Nunes’ cross.
Midfielder Rodri was shown a straight red card soon after the restart for squaring up to Morgan Gibbs-White and putting his hands on the throat of the Forest midfielder.
The Spaniard’s three-match suspension is set to add to Pep Guardiola’s selection problems, with the City boss only able to name eight substitutes – two of them goalkeepers.
City saw out the closing stages to make it six straight Premier League wins and move five points clear with their rivals all not in action until Sunday.
Luton fought back to earn a 1-1 draw against 10-man Wolves and pick up a first point in the Premier League.
Wolves had midfielder Jean-Ricner Bellegarde sent off in the 38th minute for kicking out at Luton captain Tom Lockyer after sliding in for a tackle.
Pedro Neto gave the visitors the lead five minutes into the second half after latching onto a long ball.
The Hatters, though, were back on level terms in the 65th minute when Carlton Morris converted a penalty following handball from Joao Gomes.
Luton thought they have scored a late winner through Chiedozie Ogbene, but he was flagged offside.
Crystal Palace and Fulham played out a goalless draw at Selhurst Park.
Raul Jimenez went close for the visitors late in the first half when he forced Sam Johnstone into a good save.
Willian spurned a great chance late on, but sent his low shot too close to the Eagles keeper.
Everton eased the pressure on manager Sean Dyche after securing a first Premier League win of the season with a 3-1 victory at Brentford.
Abdoulaye Doucoure gave the Toffees an early lead when he swept the ball into the top corner after a knockdown from James Tarkowski in the sixth minute and then saw a shot come back off the crossbar.
Brentford were then level in the 28th minute when Mathias Jensen converted Vitaly Janelt’s slide-rule pass – with the goal eventually ruled onside following a lengthy VAR check.
Tarkowski’s header put Everton back in front midway through the second half before substitute Dominic Calvert-Lewin wrapped things up with 20 minutes left.
Manchester United ended a run of three straight defeats in all competitions as they beat Burnley 1-0 at Turf Moor to move up to eighth place.
The Clarets went close to an early opener when Zeki Amdouni’s header was saved before his angled strike came back off the post.
United – beaten at Bayern Munich in the Champions League on the back of defeats by Arsenal and Brighton – thought they had taken the lead in the 25th minute when Jonny Evans headed in from a corner.
However, the goal was ruled out following a VAR review with Rasmus Hojlund blocking Burnley keeper James Trafford.
The visitors were, though, in front during stoppage-time at the end of the first half when Bruno Fernandes dispatched a brilliant volley into the far corner after running onto Evans’ lofted pass.
Burnley fought to get back in the game in the second half, with Sander Berge heading over from a late corner, but could not find an equaliser and now sit bottom of the table on goal difference.