Ryan Christie scored the only goal of the game as Celtic edged past Aberdeen 1-0 at Hampden Park on Sunday.
Celtic had also won by the same scoreline away to Rosenborg to keep its Europa League qualification hopes alive on Friday (AEDT), although rival Rangers have leapfrogged over it in the Scottish Premiership table.
Rodgers has won all seven domestic trophies Celtic have competed in since he took charge in 2016, with the former Liverpool boss delighted with the success following a tough start to the campaign.
"We played against a tough team at the end of a tough week," Rodgers told BBC Sport. "Probably of all the trophies that we've won, this is the most satisfying because it wasn't an easy start to the season for us.
"The players showed that mentality and strength and talent. Big hearts they've got and now another trophy."
Christie came back to haunt former club Aberdeen, where he spent last season on loan, by scoring the winner late in the first half.
"I said to him that's the football gods," added Rodgers. "Aberdeen played a big part in his development.
"He's come back and shown he's ready to be a Celtic performer and, the last two months, he's been amazing. He deserves the man of the match.
"When I first came up here, he used to be Charlie Christie's son and now Charlie is known as Ryan Christie's dad."