The 32-year-old centre-back joined Celtic for around £1 million this week as Rodgers bolstered his defensive options ahead of a hectic second half of the campaign.
The Hoops, who have a Europa League tie with Zenit St Petersburg in the new year, have signed an experience defender who has one Germany cap and over 250 Bundesliga games.
Rodgers lost one seasoned centre-back last summer when Toure hung up his boots after helping Celtic to win an unprecedented treble last term.
And the Northern Irish manager claims that Compper has the talent and experience to finally replace the former Premier League champion.
He told The Daily Record: “It was that type we were after. With all due respect to us up here when looking at centre halves you are either getting a player at the beginning of their career or one towards the end. The 32-36 bracket.
“When Kolo moved on I felt we were missing someone of that status. Marvin brings that and has history of playing at the top end of the Bundesliga.
“You have a lot of younger players here. Hopefully they improve, you gain a hefty price for them and they move on. We understand that. But I believe you need experience alongside them aswell, particularly in defensive positions.”