All Whites striker Chris Wood played the last 10 minutes as Burnley came close to earning a point at Wembley, with Tottenham needing a 91st-minute goal from substitute Christian Eriksen to win 1-0.
Crystal Palace topped Leicester 1-0, West Ham won at bottom club Fulham 2-0, Wolverhampton eased past Bournemouth 2-0, Newcastle defeated Huddersfield 1-0 and Watford beat Cardiff 3-2.
Manchester City took the lead against Everton when Jesus pounced on a through pass from Leroy Sane to slide the ball under England goalkeeper Jordan Pickford midway through the first half.
The Brazil forward made it 2-0 with a powerful close-range header from another Sane cross in the 50th minute but Dominic Calvert-Lewin gave Everton hope of mounting a comeback when he nodded in a cross from Lucas Digne 15 minutes later.
Substitute Sterling quickly restored City's two-goal advantage with his first touch of the ball, heading in a cross from Brazilian midfielder Fernandinho in the 69th minute to earn his side a 10th straight league home win.
Jesus, who signed for City in 2017, has recently been joined in Manchester by his mother.
"My confidence is back because my family is here with me," he said. "They help me a lot. It is easy for me because they are everything."
Burnley proved almost as difficult an opponent for Spurs as Barcelona did in the Champions League in midweek before Eriksen's last-gasp goal clinched victory for the North London club.
Spurs looked like enduring a disappointing comedown from their impressive 1-1 draw in Spain, but Eriksen won the game in the first minute of time added on when he coolly clipped the ball home after being teed up by England captain Harry Kane.
Crystal Palace showed they can cope without Wilfried Zaha, the club's most influential player, when the London team edged Leicester.
The Ivory Coast forward was sidelined through suspension but captain Luka Milivojevic produced an outstanding first-half finish to lift his side to 15th place.
An early goal from Raul Jimenez and a late strike by substitute Ivan Cavaleiro guided Wolverhampton past Bournemouth.
Following wins against Chelsea and Newcastle in the past 10 days, Wolves have recorded three consecutive victories in the top division for the first time since March 1980.
West Ham roared to a fourth straight league win after goals by Robert Snodgrass and Michail Antonio sank Fulham. West Ham last won four in a row in February 2014.
Newcastle ended a sequence of four matches without a victory after Venezuelan Salomon Rondon scored the only goal of the game at fellow strugglers Huddersfield Town, while Watford withstood a late rally from Cardiff City to prevail 3-2 and end a six-match winless run in the league.
- AP
