The England left-back was making just his third appearance in the competition, but emerged with a winner's medal to add to his extensive collection after Branislav Ivanovic's stoppage-time header sealed a 2-1 victory over Benfica.
It has been a successful week for the Blues, who last Saturday won at Aston Villa to all but secure a return to the Champions League they won in 2012 ahead of Sunday's final game of the Barclays Premier League season at home to Everton.
"I'm not too sure how many (medals) I've won, but I still enjoy the feeling of winning," Cole told Chelsea TV.
"I think we've got that mentality never to give up. You look at the weekend against Villa - we never gave up, kept the pressure on.
"We dominated clear-cut chances and we're lucky we've got Brana with his big head to score the header."
The victory saw Chelsea win a trophy at the eighth attempt this season and it was an unexpected success, given they tumbled into the Europa League only after suffering the ignominy of becoming the first holders to exit the Champions League at the group stages.
Cole was determined to savour the triumph and insisted it was a squad effort, with everyone playing their part, including the likes of injured captain John Terry, forward Eden Hazard and Cole's left-back deputy Ryan Bertrand.
"I'm just happy to be playing," the 32-year-old said.
"It was a pity for the people injured, like JT and Ryan, Eden Hazard. Disappointed for them. But it was a team effort through not just this game but the other games so we'll celebrate as a team."
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- Branislav Ivanovic