The 26-year-old, who has signed a two-year contract, goes straight into the squad to face Siroki Brijeg in Thursday's Europa League tie.
He made only 10 appearances for the Scottish Champions last season.
And Dons manager Derek McInnes revealed: "I met Gary before the end of last season with Celtic's permission."
The two clubs had been apart in their valuation of the player earlier this week, but an agreement was reached as Celtic themselves announced the arrival of midfielder Oliver Ntcham from Manchester City.
"Gary is a player we identified as someone we wanted to work with, but like a lot of these transfers, they are difficult and problematic," McInnes told his club website.
"A brilliant addition to the squad. It is a very exciting signing for us and I can't wait to get working with him."
Mackay-Steven, who started only five games for Celtic last season, is viewed as a replacement for Republic of Ireland winger Jonny Hayes, who moved in the opposite direction for £1.3m earlier this summer, and Northern Ireland forward Niall McGinn, who rejected the offer of a new contract to join Gwangju in South Korea.
An attempt to sign Scotland international Shaun Maloney from Hull City had stalled after the 34-year-old attacking midfielder flagged up an injury problem.
"As I have spoken about often enough, there is a real need to have that natural wide player and ball carrier," said McInnes.
"We have had the benefit and the luxury of having Jonny and Niall here for the last number of years.
"To try and get like for like is always difficult, but I think what Gary has got in his favour is that confidence, that ability to go past people, the ability to play both sides as well - he can play on the left and the right.
"Both Niall and Jonny were hugely influential and we hope Gary can be something similar.
"We hope Scott Wright will also provide us with that this season. We have now got two exciting, attacking wide players."
Mackay-Steven joined Celtic for £250,000 in January 2015 after having already signed a pre-contract agreement to join the Glasgow club that summer.
The winger, who has one Scotland cap, was with Ross County, Liverpool and Fulham as a youth but made his senior debut for Airdrie United before moving to Tannadice.
"Through all my conversation with Gary, there has always been a real willingness from him to be here," said McInnes. "He sees this as the right club for him.
"It hasn't worked out for him at Celtic and I don't think there is any disgrace in that given the level of player who is ahead of him.
"But I think he comes here with a point to prove."
Mackay-Steven becomes Aberdeen's "wild card" signing ahead of the Europa League second qualifying round first leg.
He expressed his delight at join former Celtic team-mate Ryan Christie, who has started a second season-long loan from the Scottish champions, in the Pittodrie squad.
"It's been a few weeks, maybe months, where there's been interest," said Mackay-Steven. "To finally be here, I'm delighted and can't wait to get started.
"They have been improving year on year, especially under the gaffer here now."
Mackay-Steven had been preparing for Celtic's Champions League qualifier against Linfield on Friday and feels ready to instead face the visitors from Bosnia-Herzegovina.
"Why not? Let's do it," he said. "It is a massive game for the club. European football is where this club should be."
Gary Mackay-Steven has joined Aberdeen from Celtic - having been allowed to speak to their Scottish Premiership rivals before the end of last season.