Dortmund, who finished third in the Bundesliga and will compete in next season's Champions League group stage, sacked Tuchel three days after winning the German Cup on May 27.
Tuchel had a public row with club bosses over playing a Champions League game the day after a bomb attack on the team bus in April.
Bosz, a 53-year-old former Dutch international, steered Ajax to the Europa League final this season where they lost to Manchester United, and was expected to stay on before his sudden move to Germany.
"The talk with (club bosses) convinced me that this should be the next step. I was convinced we can work well together. With respect and quality," Bosz said at his official presentation.
His move came as a surprise for Dutch fans with local media reporting a difference of opinion between Bosz and his assistant and other Ajax coaching staff.
"I am almost 20 years a coach and Ajax was a year. For me this is a great opportunity and I want to look ahead, not into the past," Bosz said, refusing to comment on the reports.
He enjoys playing a similar attacking football popular in Dortmund, and finished second in the Dutch top league in his first and only season at Ajax.
"Why Peter Bosz? We are convinced the type of football his teams play is not so dissimilar to how we want our team here in Dortmund to play," Dortmund sports director Michael Zorc said.
"Attacking football, fresh, going forward. But with good possession football and counter-pressing."
(Reporting by Karolos Grohmann; editing by Mark Heinrich)
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