"The football federation's executive board unanimously voted to put Prosinecki in charge ahead of the only other ahead candidate, Amar Osim," the body's vice-president Milorad Sofrenic told a news conference.
"We have elected a new coach after three-month long deliberations and we felt Prosinecki was the best solution because ... he has a wealth of experience in international football as a coach."
Bosnian Osim is a former player who has managed in Bosnia and Qatar.
Prosinecki, who reached the 1998 World Cup semi-finals with Croatia as a player, was the country's assistant coach from 2006 to 2010 before a 2014-2017 stint as Azerbaijan's head coach.
He took over at Bosnia from former midfielder Mehmed Bazdarevic who failed to qualify for either Euro 2016 or next year's World Cup finals in Russia.
The NFSBIH made it clear they expected the 48-year old, whose playing career included stints at Real Madrid and Barcelona after winning the 1991 European Cup with Red Star Belgrade, to turn around Bosnia's fortunes.
"We wish him the best of luck and results which have eluded us in the qualifying campaigns for the last two major tournaments," said Sofrenic.
Bosnia had reached the 2014 World Cup finals, their first major tournament as an independent nation, but made a group stage exit in the 32-nation tournament in Brazil.
(Writing by Zoran Milosavljevic in Belgrade; Editing by Jeremy Gaunt)
