Koch, who was appointed by Pope Benedict XVI as the new president of the Pontifical Council for Promoting Christian Unity (PCPCU), has been a member of the Catholic Church’s ecumenical arm since 2002.
The Archbishop of Canterbury, Dr Rowan Williams, paid tribute to Cardinal Kasper.
"Anglicans have always been made welcome by Cardinal Kasper in Rome. We have felt ourselves to be received as brothers and sisters in a common faith and mission, notwithstanding the many challenges that arise for ecumenical dialogues in our day.
"He will be greatly missed by his many friends in the Anglican Communion, not least myself.
"We wish him every blessing in all that lies ahead, and look forward to continuing, with his successor Bishop Kurt Koch, our fruitful relationship with the Holy See.”
Bishop Koch was praised for his “openness and deep ecumenical commitment” by the General Secretary of the World Council of Churches, Dr Olav Fykse Tveit.
"We rejoice at the appointment of Bishop Kurt Koch as president of the Pontifical Council for Promoting Christian Unity," he said.
"We welcome his appointment and look forward to working with him for the visible unity of the church."
Since its creation, the PCPCU has established a “cordial cooperation” with the WCC, which reportedly connects 349 churches, denominations and church fellowships in more than 110 countries and territories throughout the world.
Aside from the WCC, the PCPCU has develop dialogue and collaboration with the other church bodies and global church communions, including the Orthodox Church, the Anglican Communion, the World Alliance of Reformed Churches, the World Methodist Council, and the Baptist World Alliance, among others.
The president of the PCPCU also oversees a Commission for Religious Relations with the Jews, which has been “advancing the Catholic-Jewish friendship” over the past decade, as testified by the Jewish humanitarian, human rights, and advocacy organisation B'nai B'rith International.
Kasper demonstrated an "approach of constructiveness and healing" in his engagement with Jewish leaders, the group said in a letter this week to the retiring German cardinal.
"[W]e have been privileged to know the warmth, openness and spirit that you have brought to your work," added leaders of the organisation.
Kasper, who turned 77 in March, served as president of the PCPCU from 2001.
Bishop Koch was described by WCC’s Tveit as “a reliable partner for all those involved in the ecumenical movement.”
“We … trust he will continue Cardinal Walter Kasper's emphasis on spiritual ecumenism,” Dr Tveit stated, describing the responsibility that Koch has given as "very important ... as the call for all Christians to be one comes from Jesus Christ himself."
"I wish him much joy in fulfilling this calling, and that God may give him strength," the ecumenical leader added.
In being appointed to head the PCPCU, Koch was also promoted to the position of archbishop.