Sweeney has made over 100 appearances since joining the Blues from Newport Gwent Dragons in the summer of 2008, guiding them to the EDF Energy Cup in 2009 and the Amlin Challenge Cup the following year.
"I've really enjoyed my time with the Blues, they are a fantastic club and the supporters have been outstanding," he said. "For me, it's my home region and with the young players they have coming through the ranks, the future of the club certainly looks very bright."
The 33-year-old will bring a wealth of experience with him to Sandy Park.
He made his debut for Wales in the 2003 Six Nations against England and also featured in two World Cups, winning a total of 35 caps for his country - the last of them in 2007.
Sweeney is now excited to make his mark in the Aviva Premiership.
"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," he added.
"Like a lot of people, I've watched how they have risen over the last few years and how they've made a real name for themselves, not just in the Premiership, but in Europe as well.
"It's an exciting move for me and my young family and it's a fresh challenge.
"I know it will be the first time I've played club rugby outside of Wales, but the thought of testing myself in the Premiership is something I'm really keen to do."