Welch has played for Scarlets' feeder clubs, Llanelli RFC and Llandovery RFC before establishing himself firmly in the senior squad this season with 23 games in a Scarlets jersery in 2011/12 playing both in the RaboDirect and Heineken Cup. He has made 60 Scarlets appearances with 10 points.
Welch joins West Country club, Exeter Chiefs and with it the chance to experience rugby back in his home country in England's Aviva Premiership.
Welch said: "It's been a big honour to play for the Scarlets, and since I joined it has really pushed forward my career and experience and my sincere thanks go to the coaches and the boys for three great years at Parc y Scarlets.
"I can't speak highly enough of everyone at the region; they are a great group, with plenty of up and coming talent and ambition inside the ranks. There's a real spirit, unity and set of values at the Scarlets - something which makes you very proud to be associated with the club.
"I've been handed some great opportunities in the Heineken Cup this season and just look forward now to taking my chances in the Aviva Premiership where I've always wanted to play - as I grew up watching the English league."
Head of Scarlets Rugby Nigel Davies said: "Our thanks to Damian for his contribution to Scarlets Rugby; he has made a lot of progress with us, he's an intelligent professional and has worked hard and shown great commitment to the shirt throughout taking his opportunities well.
"As we've stated previously, we are in the process of constructing our squad for next season with particular emphasis on investment in our front five and strengthening our second row options to further our ambitions on the field in the 2012/13 season.
"We acknowledged this season that the recruitment target for us is in the second row and we're clear we need to strengthen this area of our game in order to move forward next season and we're looking to bring some quality in to work alongside a talented and strong Scarlets' back row.
"That recruitment is well underway - it's important that we take the time to get it right and that we are analytical in our approach not just reactive. We will construct our squad for next season based on getting the right players for our particular environment and team make-up and what best fits and suits our future ambitions as a team in the Heineken Cup and RaboDirect Pro12 league."