The 20-year-old has already made over 50 appearances for the club including 26 this season and was rewarded by being called into senior England squad training earlier this year following impressive performances for England Saxons
"This was an easy decision for me," said Daly. "I love playing for Wasps and this is where I want to play my rugby.
"It's been a brilliant environment for me to develop in since I left school and the style of rugby we are cultivating is really enjoyable to be involved in. We're heading in a really positive direction and I hope to play a big part in that."
London Wasps Director of Rugby Dai Young said: "Elliot has been excellent since I joined the Club and he has the quality to play in a number of positions.
"He is one of the extremely talented young English players that I want to build the future of this club around.
"There's been a great buzz around the Club this week and everyone's looking forward to testing ourselves against one of the best sides in European rugby.
"Leinster have been incredibly consistent over the last few years and this is a big challenge for our developing side but one everyone is up for."