French defender Christopher Jullien scored the only goal on the hour mark when he converted a Ryan Christie cross as Neil Lennon's side claimed their 10th consecutive domestic honour.
Rangers had most of the chances but Celtic keeper Fraser Forster was in outstanding form and pulled off a number of fine saves to deny Steven Gerrard's men.
He capped a man-of-the-match display by keeping out Alfredo Morelos' 63rd-minute penalty after Jeremie Frimpong had been sent off for bringing down the Colombian striker.
This was Celtic's 19th League Cup success overall and they are the first team to clinch the silverware four years running since the 1960s, when Jock Stein led the Hoops to five in a row.
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