Cristiano Ronaldo scored Manchester United's winner against Norwich
Mohamed Salah scored the only goal as Liverpool spoiled Steven Gerrard's return to Anfield as Aston Villa manager as the Reds, Manchester City, Chelsea and Manchester United all needed penalties to win on Saturday.
City edged 10-man Wolves 1-0 thanks to Raheem Sterling's controversial winner, Chelsea required two Jorginho spot-kicks to see off Leeds 3-2 and Cristiano Ronaldo made sure United did not slip up with a 1-0 win at bottom-of-the-table Norwich.
Gerrard was given a subdued reception at Anfield as the former Liverpool captain faced his old side for the first time in his playing or coaching career.
"I had to accept it was going to be emotional for me and my family," said Gerrard. "For me, it was trying to control that, trying to control the outside noise and be respectful to Aston Villa."
Villa were largely outplayed, but stayed in the game thanks to a combination of dogged defending, good goalkeeping from Emiliano Martinez and a couple of calls in their favour from referee Stuart Attwell, who dismissed two first-half appeals for a Liverpool penalty.
However, Gerrard's men did not escape for a third time when Tyrone Mings pulled down Salah inside the box 25 minutes from time.
Salah smashed home his 21st goal in 22 games this season to ensure Liverpool remain just one point behind City at the top of the table.
"The first 75 minutes was incredible football," said Liverpool boss Jurgen Klopp. "We played in the spaces we wanted to play but we did not finish the situations which is tricky."
City had fortune on their side to see off Wolves thanks to Sterling's 100th Premier League goal.
Wolves boss Bruno Lage was furious as Raul Jimenez was dismissed for two quickfire yellow cards before half-time.
The Mexican was booked for a trip of Rodrigo and then sent off for preventing the Spaniard taking a quick free-kick.
"The first yellow card is a crazy decision for me, in my opinion. A hard decision to accept," said Lage.
The visitors' anger was compounded by the origin of the winning goal as Joao Moutinho was judged to have handled Bernardo Silva's cross despite the ball appearing to come off the Portuguese midfielder's body onto his upper arm.
Sterling showed no mercy as he coolly slotted home from the spot to bring up his century of Premier League goals.
Chelsea needed late drama at Stamford Bridge to prevent their title challenge stumbling for the fourth time in six games.
Leeds took the lead through a penalty of their own, converted by Raphinha.
Mason Mount levelled for the European champions and order seemed to have been restored when Jorginho slotted home the first of his two spot-kicks just before the hour mark.
However, an injury-ravaged Leeds hit back once more when teenager Joe Gelhardt came off the bench to level with his first touch seven minutes from time.
But four minutes into stoppage time a third penalty of the game was awarded for Mateusz Klich's challenge on Antonio Rudiger.
Jorginho has missed a number of high-profile penalties for Italy of late, but is yet to miss for his club this season and converted to keep Chelsea just two points off the top.
"We needed this win desperately, for the belief, for the mood and for the atmosphere," said Chelsea manager Thomas Tuchel.
United were also bailed out by Ronaldo's penalty 15 minutes from time and a string of saves from David de Gea to maintain Ralf Rangnick's 100 percent start as interim manager in the Premier League.
The Red Devils moved level on points with fourth-placed West Ham after riding their luck at Carrow Road as De Gea denied Teemu Pukki and Ozan Kabak either side of Ronaldo's thumping finish from the spot.
Arsenal are one point behind United after shrugging off the absence of captain Pierre-Emerick Aubameyang due to a disciplinary issue to ease past Southampton 3-0.
Aubameyang was not included in Mikel Arteta's squad after reportedly missing training on Friday.
The Gunners comfortably bounced back to winning ways thanks to goals from Alexandre Lacazette, Martin Odegaard and Gabriel.