The visit took place to strengthen the bonds of friendship between the two Churches. In particular, it marks 40 years since the Coptic Orthodox leader Pope Shenouda III and Catholic head Pope Paul VI signed the Christological agreement between their Churches at the Vatican in 1973.
Pope Francis said the meeting between their respective predecessors united them "in an embrace of peace and fraternity, after centuries of mutual distance".
The agreement helped institute a joint commission of theological dialogue between the two Churches which "has yielded good results and has prepared the ground for a broader dialogue between the Catholic Church and the entire family of Oriental Orthodox Churches, a dialogue that continues to bear fruit to this day".
In the agreement, the two Churches professed "one faith in the One Triune God" and "the divinity of the Only-begotten Son of God".
Pope Francis spoke of his joy at being able to recognise one another as "united by one baptism" and spoke of his hope for the day when "in fulfilment of the Lord's desire, we will be able to drink together from the one cup".
"With the guidance of the Holy Spiritâ€"our persevering prayer, our dialogue and the will to build communion day by day in mutual love will allow us to take important further steps towards full unity," he said.
In his official address, Pope Tawadros said: "We appreciate all we have in common and need to work together to improve the relationships between our ancient Churches and prepare our people for our greater unity."
Bishop Angaelos, head of the Coptic Orthodox Church in the UK, was part of the official delegation accompanying Pope Tawadros on his visit to the Vatican.
"During this visit we have seen, in the two fathers of these ancient Apostolic Churches, a real commitment to working together for realised unity," said Bishop Angaelos, who is also Co-Chair of the Catholic-Oriental Orthodox Regional Forum in the United Kingdom.
"While centuries have kept us apart for a variety of reasons, it is clear that continued ecumenical dialogue and relations over the past decades have continued to bring us closer."
The formal meeting was followed by a private prayer service in Pope Francis' private chapel, led by Pope Tawadros and Pope Francis.