The 30-year-old former Scotland Under-21 captain will join the club in the summer on a two-year contract.
The former Watsonians and Boroughmuir player spent eight years in the English Championship with Bedford, Doncaster and Rotherham before moving to London and the Premiership last summer.
Cochrane said: "I'm an Edinburgh boy so it's great to fulfill a life-long ambition to play for the team where I was born and grew up.
"It's taken nine hard years of graft to prove my worth. I never gave up on earning a return to Edinburgh and I'm really excited to join.
"Wasps helped me prove that I was capable of playing at that level, so when Edinburgh came knocking at the door it was a dream come true."
Edinburgh head coach Alan Solomons added: "What's important here is that we have a good Scottish player who has gone away and done it the tough, hard way.
"We've kept a very close eye on Neil, who's got talent.
"It's great to bring a player who's from the region back into the fold. He's exceptionally committed to coming here.
"A guy like that will be massively motivated and I think will add a lot of value to our side."
- Sports & Recreation