The 28-year-old English midfielder looked to be on his way to Pittodrie in January 2016, but the deal fell through as the clubs fell out over a fee.
But he signed a pre-contract in March to join the Dons on a three-year deal.
"I always wanted to come here and I'm just delighted it has eventually been done," he told the Dons website.
"What's happened in the past has happened in the past, but it's been a long wait."
Tansey had returned to the Caledonian Stadium for a second spell at the club after leaving Stevenage in 2014.
He scored nine goals in 44 appearances this season but was unable to prevent Caley Thistle being relegated.
Tansey becomes the latest Caley Thistle player to have ended up at the other end of the A96.
Dons striker Adam Rooney and midfielders Jonny Hayes and Graeme Shinnie all moved to the Granite City having previously played for Inverness.
"I always had it in my mind to come here because of the set-up and from what the lads I know have said about the place," added Tansey. "It spoke volumes about the place.
"It was pretty much a no-brainer to come here.
"I've spoken to a few of the boys here - Graeme Shinnie, Jonny Hayes, Miles Storey, Adam Rooney and I've played against them over the years.
"They're all great lads and top players and it is exciting to play with players like that.
"That's what you want to do. It brings you on as a player and puts pressure on you to deliver for them as well as yourself."
Greg Tansey is relieved to have ended his long wait to sign for Aberdeen after leaving relegated Inverness Caledonian Thistle.