The 22-year-old, who has scored nine goals in 16 games during a hugely successful loan spell with the Glovers, has agreed a two-and-a-half year contract and moves for an undisclosed fee.
"Obviously I am delighted to sign permanently at Yeovil Town," he told the club's official website. "I have really enjoyed my loan spell here and I am looking forward to scoring more goals for the club."
Manager Gary Johnson added: "I am very pleased that Paddy has joined us on a permanent contract. He has been fantastic for us since joining on loan and will add to the quality we already have in the squad.
"The directors, management, players and fans all wanted Paddy to stay with us and I am delighted to have secured his signature."
Madden's move was announced just hours after his double helped Yeovil record a 3-0 win over Leyton Orient at Huish Park.
The Irishman moved to Carlisle from Bohemians 12 months ago but made just eight starts for the Cumbrians before being loaned out to Somerset.