Sana, who made his debut for Sweden in a thrilling 4-4 draw away to Germany in 2012, spent two and a half years in Amsterdam after leaving IFK Gothenburg, but failed to make the breakthrough in Dutch football.
He joins a Malmo side that won their second successive Swedish title in 2014 and are looking to qualify for the group stages of the Champions League again next season, having faced Juventus, Atletico Madrid and Olympiakos in this year's competition.
"This is what I have been longing for, to play in front of people who are committed and who have a big interest," Sana said in a statement.
Malmo sporting director Daniel Andersson added: "He made a big breakthrough and came up to a high level where he was selected for the national team.
"He has had a tough time since then but we know the capacity that Tobias has and we hope to be able to bring out all the footballing talent he has within him."
(Reporting By Philip O'Connor, editing by Alan Baldwin)
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