The popular tighthead, whose contract came to an end in 2012/13, made his debut for the Scarlets against Benetton Treviso in September 2010 and his first Scarlets try came in a LV= Cup match against Sale Sharks in February 2012.
Edwards has also played for Newtown, North Wales, Llanelli RFC and Llandovery.
He said: "I'd like to say a big thank you to everyone at the Scarlets. I have really enjoyed the last three years and will always be a big Scarlets fan having lived in the area and having my young family there.
"I have made some great and close friends at Parc y Scarlets - not only the guys in the squad but also in our rugby community in West Wales, particularly our supporters who have been brilliant and I really appreciate all the support I have had.
"It is always hard to move on but this gives me a new and fresh opportunity with a new group of players and a lot of new faces joining London Welsh plus hopefully the chance to get a bit more rugby every week.
"It has been a massive honour to have been part of the Scarlets - there are not many international sides let alone club sides that have beaten the All Blacks and to be a small element of that history and to have worn that jersey is something I will always be very proud of.
"I'd also like to express my thanks to those involved at Llandovery who are doing a great job at developing and supporting rugby players and helping bring forward more Welsh talent.
"I'll be keeping touch with everyone at Parc y Scarlets and will look forward to seeing the team do well next season."
Scarlets Head Coach Simon Easterby said: "It's not easy seeing players move on from your environment and Pete has been a popular member of the squad and a loyal Scarlet throughout his time with us.
"We thank him for what he has given to our region and wish him all the best in his new contract with London Welsh in the season ahead."