Chelsea leave it late as Cole sees off Stoke
The European champions, without a win in their last three matches in all competitions, moved on to 13 points from five games ahead of Everton and West Bromwich Albion who have 10.
Everton played superbly to win 3-0 at Swansea City while West Brom won their third successive home match from the start of the season for the first time since 1954 with a 1-0 win over promoted Reading who dropped to bottom place.
Fulham, who won 2-1 at Wigan Athletic moved up to fourth on nine points.
Promoted Southampton climbed off the bottom of the table as they collected their first points of the season with a come-from-behind 4-1 win over a ragged-looking Aston Villa.
Chelsea's Cole, who had not scored in the league for over two years, struck with five minutes to play at Stamford Bridge when he hammered the ball past Stoke goalkeeper Asmir Begovic.
Chelsea manager Roberto Di Matteo started with skipper John Terry and Frank Lampard on the bench, bringing on Terry for the final few minutes as Stoke came surging forward in search of what would have been a deserved point.
Cole said: "It was a battle. Stoke are always physical and make it hard for us to play our football.
"They defend deep with ten men but for our goal it was a great move and I took a chance to get into the box. It is mentally good to get a lead in the table and last season we didn't have a good time in the Premier League too often."
Everton beat Swansea with goals from Victor Anichebe, Kevin Mirallas and Marouane Fellaini in an exciting lunchtime match in Wales in which both sides also hit the bar.
Swansea were reduced to 10 men when Nathan Dyer was sent off 12 minutes after coming on as a half-time substitute but even with 11 men they had been no match for a far superior Everton.
West Brom moved into third place after Romelu Lukaku - on loan from Chelsea - scored what proved to be the winner after 71 minutes at The Hawthorns.
The biggest turnaround came at St Mary's where Southampton, without a point after four matches, trailed 1-0 to Aston Villa whose England striker Darren Bent gave the visitors the lead after 36 minutes with his first goal of the season.
Saints came storming back with Rickie Lambert equalising after 58 minutes and defender Nathaniel Clyne, who conceded an own goal against Arsenal last week, scoring in the right net for the first time since a goal for Crystal Palace three years ago.
He struck five minutes after Lambert's opener with a fine shot after starting and finishing the move himself.
Southampton kept marching forward with Jason Puncheon adding a third and Lambert completing the victory with a 90th minute penalty as they moved out of the relegation zone and up to 17th.
Coach Nigel Adkins said: "It was a good three points for us. We have had a winning mentality for the last few years but the Premier League is a massive challenge for everybody and we are sticking together and today was a well-deserved three points."
Fulham continued their promising start with a 2-1 win at Wigan Athletic where Hugo Rodallega scored the opener against his former team and Damien Duff adding the second before the home side grabbed a late consolation through Arouna Kone.
Sunderland were close to ending a run of 11 games without a win dating back to last March when they led West Ham United for most of the match at Upton Park after a brilliantly taken left-foot shot from Steven Fletcher after nine minutes.
But West Ham captain Kevin Nolan equalised in stoppage time to rescue a point for the Hammers with Sunderland now having drawn all four of their league matches so far this season.