The 25-year-old made 13 appearances for Welsh in this season's Premiership, and scored a try when they claimed their first victory in English rugby's top flight by beating Exeter last September.
Before joining the Exiles, Jackson played for Bath and Doncaster.
"Ed is a player I am excited about," Wasps rugby director David Young said.
"He was excellent last season, and he has backed it up this year again in the Premiership.
"He is an explosive runner with great hands, and he can play eight and six, which will be very useful for us."