Everton suffered a shock 3-2 home defeat by mid-table Crystal Palace at Goodison Park to put a huge dent in their bid for a top-four finish.
City, champions in 2012 and runners-up to neighbours Manchester United last season, failed to respond to their 3-2 defeat at leaders Liverpool on Sunday and stayed third in the table on 71 points from 33 matches, six behind Liverpool and four adrift of Chelsea with only one game in hand.
Liverpool are in pole position to claim their first English title in 24 years, but they still have to play Chelsea at Anfield on April 27.
"We did not play well but we had in our mind the game against Liverpool," City manager Manuel Pellegrini told Sky Sports.
"It is not enough because we have Liverpool and Chelsea on top of the table and we needed to add three points - will see in the future what will happen but our chances are less."
Sunderland remained rooted to the bottom of the table with 26 points from 33 matches, six points adrift of 17th-placed Norwich City.
"We deserved better," manager Gus Poyet said. "We conceded a goal after two minutes at Manchester City and every single person in the world expected us to lose from there.
"But we passed the ball and created chances. We kept believing."
After the damaging defeat by Liverpool, City knew anything other than a win would seriously dent their title hopes and a slick passing move enabled Fernandinho to fire them ahead after two minutes.
But with talismanic midfielder Yaya Toure missing through injury, City lacked a cutting edge and Sunderland striker Connor Wickham was in the right place on 73 minutes to convert Emanuele Giaccherini's left-wing cross and score his first Premier League goal since 2011.
City have won just twice in their last eight games against Sunderland and Wickham put Poyet's side in front after 83 minutes when he beat Joe Hart at his near post.
But Sunderland's hopes of a shock victory were dashed when goalkeeper Vito Mannone spilled Nasri's shot into his own net two minutes from time.
Goals from Jason Puncheon and Scott Dann gave Crystal Palace a 2-0 lead at Goodison Park before substitute Steven Naismith pulled a goal back for Everton.
Cameron Jerome's strike restored Palace's two-goal advantage and Kevin Mirallas grabbed a late consolation for Roberto Martinez's side.
Everton's defeat, which followed a run of seven wins in a row, left them on 66 points from 34 matches, a point adrift of fourth-placed Arsenal with four games remaining.
"It's a real disappointment, a result which is hurtful," said Martinez.
"The first half we were fearful, the way we conceded the first goal shows that. They didn't have the threat to score three goals."
Victory for Tony Pulis's Palace continued their surge up the table and they are in 11th place with 40 points.
"It was an excellent performance from us today," Pulis told the BBC.
"To get 40 points is a fantastic achievement for this group of players. But we owe it to the Premier League to remain competitive until the end of the season."
Palace, seeking their fourth consecutive league win, took the lead after 23 minutes when Yannick Bolasie escaped from left back Leighton Baines and put a cross into the box which goalkeeper Tim Howard failed to deal with and Puncheon applied a clinical finish.
Winger Bolasie gave Everton another scare when he smashed a 20-metre strike against the base of the post, but Palace extended their advantage on 49 minutes when Dann headed in from a corner.
Martinez introduced Naismith in the hope of clawing a goal back and it worked as the Scottish striker poked the ball home on 61 minutes.
Jerome restored Palace's two-goal cushion before Belgium winger Mirallas pulled a goal back for Everton when he clipped the ball past Julian Speroni to set up a grandstand finish but the visitors held on.