George Boyd clinched a 1-0 victory for relegation-threatened Burnley after meeting City captain Vincent Kompany's clearance from Kieran Trippier's free kick with a half-volley in the 61st minute.
City, which has lost two of its last three league games, remains five points behind Chelsea with nine games to go. The leaders have two games in hand on the champions, and host Southampton on Sunday.
"Of course we are doing something wrong because we are not winning the games that we normally do against these kinds of teams (Burnley)," City manager Manuel Pellegrini said. "Maybe the chance of the title is now far but we have to continue fighting for our position."
City's domestic difficulties come ahead of a trip to Barcelona where Pellegrini's side will be trying to overturn a 2-1 deficit in the Champions League round of 16 on Wednesday.
Arsenal prepared well, though, for its trip to Monaco to turn around a 3-1 first-leg loss by easing to a 3-0 victory over West Ham.
Olivier Giroud inspired Arsenal's fifth straight league victory by scoring one and setting up Aaron Ramsey's goal at Emirates Stadium. Mathieu Flamini also netted as Arsene Wenger's team tightened its grip on third place and moved within a point of Man City.
Aston Villa ended its barren scoring run away from home in spectacular style at Sunderland.
Villa had taken seven months to score four away goals in this season's league campaign, only for the team to double that amount in the first 45 minutes in a 4-0 win at Sunderland. Christian Benteke and Gabriel Agbonlahor both scored twice.
The glut of goals sparked unrest and a mini-exodus among home fans at halftime, with some supporters confronting Sunderland manager Gus Poyet close to the dug-out. With 10 minutes left, the Stadium of Light appeared half-empty.
"When you are a fan and you see your team performing like we did in the first half, I would do the same," said the under-pressure Poyet, who described the display as "unacceptable."
Crystal Palace beat Queens Park Rangers 3-1 at home in a match marked by an outrageous 40-meter consolation strike by Matt Phillips for the visitors. West Bromwich Albion overcame Stoke 1-0 and Leicester drew 0-0 with relegation rival Hull.
Arsenal will head to Monaco increasingly confident of securing yet another season in Europe's top competition by finishing in the top four.
The Gunners are fighting with Manchester United, Liverpool, Tottenham and Southampton for the two spots behind the current top two of Chelsea and City, and their form suggests they will be difficult to stop.
Giroud set Arsenal on its way to an eighth straight home win by smashing in a left-foot shot off the post in first-half injury time. The France striker provided the square ball that Ramsey steered home in the 81st before Flamini added a third in the 84th.
West Ham is winless in seven league games as it drifts toward mid-table after challenging for the top four in the first half of the season.
Villa hadn't scored an away goal in more than 10 hours before Benteke's 16th-minute goal against Sunderland. Agbonlahor scored in the 18th and 37th minutes before Benteke piled on the agony for the hosts with a fourth in scarcely believable scenes.
Villa has been revived by new manager Tim Sherwood and a second straight win in the league — after seven defeats in a row — pushed the team above Sunderland further clear of relegation.
Sunderland has won one of its last 11 league games at home.
QPR stayed in the relegation zone after another pitiful away performance, in which it conceded three goals in the first half to Wilfrid Zaha, James MacArthur and Joel Ward.
Phillips' late consolation was one of the goals of the season in the Premier League, a dripping strike from long range with flew in off the underside of the crossbar.
Brown Ideye scored West Brom's winner to move the team into mid-table and continue its resurgence under new manager Tony Pulis.Suggest a correction