Champions Arsenal, helped by a double from Sweden midfielder Fredrik Ljungberg, kept Chelsea's lead at five points with a 3-1 success at Charlton Athletic while Manchester United, who won 2-0 at Middlesbrough in an evening kickoff, are four points further back in third.
England midfielder Cole's low drive from distance deflected in off Jamie Carragher 10 minutes from time to hand Jose Mourinho's Chelsea their third win in the hectic holiday programme and a perfect start to their centenary year.
"I think we had that bit of luck that normally champions need to have at crucial moments," Mourinho told Sky Sports.
Chelsea have 52 points from 21 games, clear of Arsenal on 47 and Manchester United on 43.
Fourth-placed Everton succumbed to their second successive defeat, a 5-2 drubbing at resurgent Tottenham Hotspur. All five of their league defeats this season have come against sides from the capital.
Cole, scoring for the second successive game, had also hit the winner in Chelsea's 1-0 victory over Liverpool at Stamford Bridge in October.
On as a 76th-minute replacement for Damien Duff, Cole was well positioned to latch on to Glen Johnson's cushioned header from Arjen Robben's corner.
DEFENSIVE QUALITIES
Mourinho praised Chelsea's defensive qualities but said his side, who have lost only once in their 21 league matches and are unbeaten in their last 12, had not been at their best at Anfield.
"It was a fantastic defensive performance, not just by the team but by some individuals. We were a bit lucky," he said.
"The way both teams fought and the way both teams tried to win, the draw would have been a fair result."
Liverpool were incensed when they were refused a clear penalty late in the first half after Portugal midfielder Tiago handled from a corner.
The former European champions suffered a further blow when it was confirmed that Spanish midfielder Xabi Alonso had sustained a broken ankle after a strong tackle from Frank Lampard and will be out for six weeks.
Arsenal manager Arsene Wenger praised his side's resilience for carving out victory against Charlton at The Valley where they were held 1-1 in their march to the title last season.
"We needed to dig deep and produce a great performance to come home with three points," he said. "All our experienced players were outstanding today."
Ljungberg turned sharply to fire the champions in front on 35 minutes but his goal was cancelled out by a thunderous free kick on the stroke of halftime by Moroccan defender Talal El Karkouri. Ljungberg finished off a slick move to put Arsenal back in front and Dutchman Robin van Persie added a third.
GIGGS AGAIN
Ryan Giggs scored for the third successive match as Manchester United made it eight wins from nine games by easing past fifth-placed Middlesbrough at the Riverside.
The Wales international produced a deft finish 11 minutes from time to seal the points. Alex Ferguson's side had led early on when midfielder Darren Fletcher tapped into an empty net after keeper Mark Schwarzer parried a Giggs shot into his path.
Spurs have never lost to Everton in the Premier League and that record was never threatened as the Merseyside club had their Champions League pretensions severely questioned at White Hart Lane. Dean Marney (two), Reto Ziegler and Pedro Mendes netted their first goals for Spurs.
At the bottom, West Bromwich Albion were five minutes away from celebrating their first win under Bryan Robson but El Hadji Diouf's equaliser earned Bolton Wanderers a 1-1 draw.
Norwich moved out of the bottom three after a 1-1 draw at Portsmouth, leapfrogging Crystal Palace who lost 3-1 at Fulham.
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