Sterling sends City eight points clear, Sanchez lifts Arsenal
City have swept aside all comers this season, but they found themselves in the unusual position of trailing at half-time after Nicolas Otamendi's own goal at the John Smith's Stadium.
Pep Guardiola's side quickly equalised through a Sergio Aguero penalty in the second half before Sterling's lucky 84th-minute winner sealed their 11th successive league victory.
City's gritty success reopened an eight-point advantage over second-placed Manchester United, who had temporarily closed the gap with a 1-0 win against Brighton and Hove Albion on Saturday.
The leaders had controlled the first half yet found themselves behind in stoppage-time when a Tom Ince corner hit Otamendi on the shoulder and flashed into the net.
Guardiola's men gave an impressive response to that shock setback as they equalised two minutes after the interval.
Sterling was fouled by Scott Malone and Aguero drilled in the spot-kick for his 11th goal of the season.
City pressed for a winner, with Leroy Sane hitting the bar and Fernandinho booked for diving in the penalty area after a challenge from Rajiv van La Parra.
They finally got the victory their dominance deserved late on, but the goal was fortunate in the extreme.
Huddersfield goalkeeper Jonas Lossl blocked from City's Gabriel Jesus inside the penalty area.
But the rebound hit Sterling and bounced over the line for his 12th goal of the season -- and his second winner this week after his decisive strike against Feyenoord in the Champions League.
Arsenal's Chilean striker Alexis Sanchez celebrates scoring a penalty during the English Premier League match between Burnley and Arsenal at Turf Moor in Burnley on November 26, 2017
At Turf Moor, Arsene Wenger's side looked set for a frustrating draw as they entered stoppage-time, but they snatched the points when James Tarkowski was ruled to have pushed Aaron Ramsey.
Sanchez held his nerve to slot home the dramatic late penalty and moments later Arsenal were celebrating a 1-0 win that lifted them into fourth place in the Premier League.
"From outside it (the penalty) looked 100 percent. I don't know why he pushed him in the back. I would like to see it again," Wenger said.
"In the first half we didn't find our game. Burnley defend extremely well and are very dangerous on the direct game."
Burnley boss Sean Dyche fumed: "There's a lot of me inside that's raging a lot.
"It was highly unlikely it wasn't going to get given on a day like today, that's all I'll say (about the penalty)."
Southampton's Danish midfielder Pierre-Emile Hojbjerg (right) vies with Everton's Icelandic midfielder Gylfi Sigurdsson during their English Premier League match in Southampton on November 26, 2017
Without Mesut Ozil, who was ruled out due to illness, Arsenal struggled to break down Burnley.
But Chile forward Sanchez stepped up to seal Arsenal's second successive league win with his fourth club goal of the season.
At St Mary's, Everton's miserable season hit another low point as Southampton romped to a 4-1 win -- giving caretaker boss David Unsworth a fifth defeat in his seven matches in charge following Ronald Koeman's sacking.
Everton, linked with a move for Republic of Ireland manager Martin O'Neill, are now just two points above the relegation zone.
"I cannot stand here and defend them, it was an unacceptable performance, the fans will be furious and so am I. It's not good enough," Unsworth said.
Having hit just nine goals in their previous 12 league games, Southampton took advantage of woeful Everton defending to win for the first time in four league matches.
Dusan Tadic put Saints ahead in the 18th minute when he beat Phil Jagielka to Ryan Bertrand's pass and held off the England international before guiding the ball past Everton goalkeeper Jordan Pickford.
Gylfi Sigurdsson equalised for Everton on the stroke of half-time, the Iceland midfielder cleverly working space for a superb 25-yard strike that went in off the crossbar.
But Southampton needed just seven second-half minutes to regain their advantage as Charlie Austin headed home from Bertrand's cross.
Austin repeated the feat six minutes later, arriving to nod Tadic's cross home, and Steven Davis got Southampton's fourth in the 87th minute.
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