The 33-year-old has been with the Pro12 side since 2008, having started his career at Newport Gwent Dragons.
The fly-half told the Pro12's official website: "I've really enjoyed my time with the Blues, they are a fantastic club and the supporters have been outstanding.
"I take with me a lot of happy memories and a lot of good friends, but I wanted a fresh challenge at this stage in my career and that is something Exeter have offered me.
"Exeter are a fantastic club and as soon as I heard they were interested in me, I was keen to talk to them and have a look around the place."
Sweeney has signed a one-year deal with the Chiefs.