City jumped to second place on Saturday and are three points behind Arsenal, who won 1-0 at Fulham thanks to another set-piece goal.
Liverpool can move above City when they host Manchester United on Sunday.
Haaland has 23 goals in 13 games for club and country this season. After heading in a cross from Nico O'Reilly, he met a cut-back from Savinho with a shot that went through the legs of defender James Tarkowski and past the unsighted Jordan Pickford in goal.
Jack Grealish wasn't able to play for Everton under the terms of his loan move from City.
Arsenal's winner at Fulham came from a familiar source: Bukayo Saka swung in a corner, defender Gabriel Magalhaes flicked it on and Leandro Trossard converted at the back post.
It was a sixth win in eight games for Mikel Arteta's Arsenal side.
The lunchtime match ended in a humbling 3-0 home defeat for Nottingham Forest and the sacking of Australian manager Ange Postecoglou, barely 20 minutes after the final whistle.
Josh Acheampong, Pedro Neto and Reece James scored for Chelsea, who had Malo Gusto sent off late on – Chelsea's fifth red card this season.
Jean-Philippe Mateta scored a hat-trick — including an equaliser from the penalty spot in the seventh minute of stoppage time — in a breathless 3-3 draw against Bournemouth.
Nick Woltemade's latest goal for Newcastle — a deft flick between his legs into the corner — proved to be in vain in a 2-1 loss at Brighton, who scored a goal in each half from Danny Welbeck.
Burnley beat fellow promoted side Leeds 2-0, while Sunderland won at home to Wolverhampton 2-0 to move up to seventh place in the table.
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