Cardiff-born Burns, capped 18 times by Wales at U18 level, was included in Chris Coleman's pre-Euro 2016 training squad.
Aberdeen manager Derek McInnes, who worked with the 21-year-old when in charge at Ashton Gate, told the club's website: "I am delighted to get him.
"Wes is a fantastic young talent who can play right across the front line. He has got a brilliant attitude and plenty of motivation to do well with us.
"I would like to thank Bristol City and Lee Johnson for their help in making the deal happen as Wes had plenty of options."