Backe, a 63-year-old native of Sweden, previously managed FC Copenhagen, who under his stewardship won three Danish championships, and acted as a deputy coach to Sven-Goran Eriksson at Manchester City in 2007-2008.
Backe's most recent post as a coach was at New York Red Bulls from 2010-2012.
He also served as deputy coach for Mexico in 2008-2009, but Finland will be his first stint as a head coach for a national team.
He will take up his new post at the start of 2016, the SPL said.
"Everything is possible," Backe told a news conference announcing his appointment on a two-year deal which will run from January 2016, citing the success of Iceland under compatriot Lars Lagerback.
"My job will be to lift the self-confidence of the Finnish players so that they believe that everything is possible."
Finland's previous coach Mixu Paatelainen was sacked in June, following a fourth straight defeat in the Euro 2016 qualifying campaign.
That poor run of results leaves Finland second from bottom of Group F with four points from six games and little hope of qualifying for the tournament in France.
Finland have never qualified for a World Cup or European Championship finals and are currently ranked 89th in the world.
(Reporting By Anna Ercanbrack; Editing by: Ossian Shine)
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