Stoke twice hit back to level after falling behind at Stamford Bridge, but Willian's second goal and a late Diego Costa strike saw Chelsea match Arsenal's run of 13 wins within the same season from 2001-02.
Antonio Conte's side pulled nine points clear of second-place Liverpool, who host third-place Manchester City later in the day in an explosive climax to the calendar year.
Manchester United came perilously close to seeing their own winning streak end, only for late goals by star signings Anthony Martial and Paul Pogba to secure a 2-1 home win over Middlesbrough.
It was sixth-place United's fifth straight league win and took them to within a point of fourth-place Arsenal, who host Sam Allardyce's Crystal Palace on New Year's Day.
Chelsea welcomed back N'Golo Kante and Costa from suspension for the visit of Stoke and took a 33rd-minute lead when Gary Cahill headed in a corner from Cesc Fabregas.
Bruno Martins Indi equalised within seconds of kick-off in the second half, slotting in after Peter Crouch nodded down Charlie Adam's free-kick.
After Willian had restored the hosts' lead, lashing in from Eden Hazard's lay-off, Crouch drew Stoke level again in the 64th minute when he tapped in Mame Biram Diouf's cross.
But Willian put Chelsea back in front a minute later and Costa's late strike sealed victory.
At Old Trafford, United produced a comeback reminiscent of the glory years of the watching Alex Ferguson to continue their recent upturn.
But they had to do it the hard way, with Ibrahimovic initially denied an 18th goal of the season in a contentious incident late in the first half.
Leaping to meet Anthony Martial's left-wing cross, the Swede jabbed the ball past goalkeeper Victor Valdes at the near post, only for referee Lee Mason to rule it out.
United's players, and Jose Mourinho, contested the decision, and there was worse to come when Grant Leadbitter drilled Boro in front from Alvaro Negredo's knock-down in the 67th minute.
United got their goal in the 85th minute thanks to a cool finish from Martial, and barely a minute later Pogba headed in the winner from Juan Mata's cross.
Champions Leicester City claimed a much-needed win, beating West Ham United 1-0 courtesy of Islam Slimani's 20th-minute header to climb six points clear of the relegation zone.
Swansea City made a demoralising start to life without sacked manager Bob Bradley, slumping to a 3-0 home defeat against Bournemouth that kept them at the foot of the table.
Goals from Benik Afobe, Ryan Fraser and Josh King gave Bournemouth victory, with first-team coach Alan Curtis taking temporary charge of Swansea.
Andre Gray became the first Burnley player to score a Premier League hat-trick as Sean Dyche's men beat third-bottom Sunderland 3-1 to climb to 11th place.
Gray scored three times in the first 53 minutes at Turf Moor, with Ashley Barnes netting a penalty before Jermain Defoe replied for the visitors.
Matt Phillips and Wales's Euro 2016 star Hal-Robson Kanu scored as West Bromwich Albion came from behind to win 2-1 at Southampton, who had centre-back Virgil van Dijk sent off late on for two bookable offences.