Austrian coach Hutter has signed from Young Boys, who he led to the Swiss Super League title this term, on a contract until 2021.
The 48 year-old, a double winner with Red Bull Salzburg three years ago, will be joined by assistant Christian Peintinger when he replaces Kovac in Frankfurt.
"I had three wonderful years at Young Boys and was extremely happy there. I'll always remember my time there, especially our title triumph and the incredible emotions and celebrations that followed," said Hutter in a statement. "Joining Eintracht Frankfurt is a dream come true for me. I never appeared in the Bundesliga during my playing days, even though I once had an offer.
"I've turned down one or two coaching offers from the Bundesliga in the past, so I'm all the more excited to be able to work in the league of the world champions."
Hamburg opted against appointing a new manager after being relegated from the Bundesliga for the first time in its history and have instead handed Christian Titz, who took over in March, a new two-year contract.
"I feel at home in Hamburg and at HSV and have always said I could see myself here longer than the season that has just passed," said Titz. "We all know that big challenges await us in the 2.Bundesliga. We now need to give that our full focus."