Pep Guardiola’s side had watched Manchester United draw level on points with them on Tuesday after their bitter rivals crushed Southampton 9-0.
But City underlined their determination to hold onto top spot with a composed, mature display at Turf Moor.
Jesus opened the scoring after a mistake from Burnley keeper Nick Pope and Sterling grabbed the second before half-time.
After a spluttering start to the season that threatened to derail their title hopes, City, who have a game in hand on second placed United, are in pole position to dethrone champions Liverpool.
They head into Sunday’s crucial clash at Liverpool on a club record run of 13 successive victories in all competitions, including nine in the league.
Arsenal were the last English top-flight side to win 13 consecutive games in all competitions in 2002.
With Sergio Aguero and Kevin De Bruyne still sidelined by injuries, City’s surge to the top is even more remarkable.
Last crowned champions in 2019, they are unbeaten in 20 matches in all competitions stretching back to November and have conceded just four goals during that run.
It is City’s miserly defence that has been the key to their resurgence following the signing of Ruben Dias from Benfica.
Their latest rock-solid effort at the back equalled the club league record of six consecutive clean-sheets.
Burnley are 16th after a second successive defeat, but they remain eight points above the relegation zone.
– Ruthless City –
Guardiola made five changes from last weekend’s win against Sheffield United, with Sterling and John Stones back in the starting line-up.
Aguero is several weeks from returning to action after recently emerging from coronavirus quarantine, leaving Jesus to shoulder the burden of leading City’s attack.
Finally rising to that challenge, Jesus struck in the third minute with a clinical close-range header after Pope tamely parried Bernardo Silva’s shot into the Brazilian’s path.
It was a bad mistake from Pope but Jesus wasn’t complaining about the gift.
After scoring just once in 12 games, the 23-year-old, who hit the winner against Sheffield United, has now netted three times in his last four appearances.
City had never lost in the 37 previous league games that Jesus had scored in and Guardiola’s side always looked likely to extend that streak as they dominated possession.
Probing away with ruthless efficiency, City’s pressure was rewarded with a second goal in the 38th minute.
Mahrez fed Ilkay Gundogan down the right flank and the German’s superb low cross reached Sterling, who slotted home from close-range for his 10th club goal of the season.
Joao Cancelo could have killed off Burnley before half-time, but the Portugal defender shot straight at Pope from Mahrez’s pass.
Burnley didn’t have a single shot in the first half and that gulf in class remained obvious after the break.
Gundogan has made a habit of scoring eye-catching goals in recent weeks and he wasn’t far away from another as his 20-yard blast fizzed just over.
Sterling was denied by Pope’s out-stretched boot after surging clear of the Burnley defence.
Ederson escaped with a yellow card after the City keeper clattered Erik Pieters outside his area, but that was a rare scare on a flawless night for the leaders.