Dons boss Craig Brown revealed at the weekend he was hopeful of securing the services of the 34-year-old before Tuesday's transfer deadline.
And the former Cardiff man has now joined the Pittodrie club on an 18-month deal.
Aberdeen chief executive Duncan Fraser told the club's official website: "I have nothing but the highest praise for the way Dundee conducted themselves in this transfer.
"They were desperate to keep Gavin and pulled out all the stops to do so but, when it was obvious he wanted to come to Aberdeen, they were first class in making sure there were no unnecessary delays."
Rae is cup-tied for Saturday's William Hill Scottish Cup tie with Queen of the South, meaning he will have to wait until the following week to make his Dons debut.
(reopens) Rae revealed he opted for a move to the Dons despite holding talks with Clydesdale Bank Premier League rivals Hibernian.
He said: "I spoke to Hibs on Thursday and I was down at the training ground.
"But, between Hibs and Aberdeen, I was going to pick Aberdeen, because I'm from here and wanted to play for them."
He added: "Hopefully I'll bring experience to the team.
"I just want to be part of a strong squad. There are a lot of good young players here and hopefully with a good blend we can push on."
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