Cole, who last scored against Wigan Athletic in May 2010, lifted an 85th-minute winner over visiting goalkeeper Asmir Begovic after the ball had been knocked into the box by Juan Mata.
Cole's winner was tough on Stoke which had defended doggedly and had chances itself with Marc Wilson testing Petr Cech from long range and Jonathan Walters and Matthew Etherington both firing wide.
But Roberto Di Matteo's men also looked dangerous, Fernando Torres scuffing the best chance into the arms of Begovic in the first half, and Oscar twice being denied by the excellent Potters goalkeeper.
Southampton stormed back from behind to earn its first points of the season with an emphatic 4-1 triumph over ailing Aston Villa at St Mary's.
It looked like being the same old story for Southampton, which had lost its first four matches back in the top flight, when Darren Bent pounced on hesitant defending to sweep home a 36th-minute opener.
Rickie Lambert lashed home Southampton's equaliser on 58 minutes and five minutes later it was ahead when defender Nathaniel Clyne slotted past Villa goalkeeper Brad Guzan.
Saints made sure of victory on 72 minutes when Jason Puncheon's right-footed drive took a heavy deflection off Ciaran Clark, and Lambert sealed victory with a penalty in the dying seconds.
Steven Fletcher scored his fourth goal in three appearances for Sunderland but it was not enough to claim three points for the Black Cats as West Ham United snatched a late 1-1 draw at Upton Park.
Fletcher finished off a fine Sebastian Larsson cross in the ninth minute and came close to a second soon afterwards when his effort in the box was blocked by James Collins.
Sam Allardyce's men poured on the pressure with both Ricardo Vaz Te and Kevin Nolan coming close before Nolan volleyed the equaliser in injury-time from a header down by substitute Modibo Maiga.
West Bromwich Albion continued its superb start to the season as Romelu Lukaku drilled the only goal of the game in the 71st minute to see off Reading 1-0 at The Hawthorns.
Hugo Rodallega came back to haunt his former club as he netted the opener for Fulham in a 2-1 win against Wigan Athletic at the DW Stadium.
Rodallega's 31st-minute header set Fulham on the way to three points and Damien Duff slotted home to make it 2-0 on 68 minutes. Arouna Kone netted an injury-time consolation for the host.
In the early kick-off, Everton continued its promising start to the season by coasting to a 3-0 win at Swansea City.
Victor Anichebe opened the scoring on 21 minutes and Kevin Mirallas and Marouane Fellaini added further goals, while the Swans were reduced to 10 men after substitute Nathan Dyer's 58th-minute dismissal.