The attack, which happened approximately 140 miles south of Cairo, also left 22 others injured. Children were among the bus passengers.
Most Rev Justin Welby said: "I am heartbroken by the news of another awful attack on men, women and children, murdered because of their faith in Jesus Christ.
"In this time of deep sorrow and pain, we commit to prayer those who have died, those who have been injured and those who have lost loved ones.
"We pray that all might know the presence of God in this dark time and draw closer to the Great Redeemer, who is Jesus Christ."
Bishop Angaelos, the General Bishop of the Coptic Orthodox Church in the UK, told Premier's News Hour: "It was said enough seeing what happened in Manchester [in the concert attack on Monday] - a targeted on children and young people.
"This was very similar; it was a bus filled with family members who were travelling on a pilgrimage to a monastery in Minya."
The group was travelling from Beni Suef province to the remote monastery of St Samuel the Confessor in Maghagha.
While no group immediately claimed responsibility for the attack - the fourth targeting Christians since December 2016 - it bore the hallmarks of Islamic State.
Fr Rafic Greiche, a spokesman for the Egyptian Catholic church, told a local television station: "The growing number of these terror attacks is not at all reassuring."
Two suicide bomb attacks at churches in Alexandria and Tanta left 45 people dead on Palm Sunday this year. A suicide bombing at a chapel next to St Mark's Coptic Orthodox Cathedral in Cairo on 11th December 2016 killed 29 people.
President Abdel-Fattah el-Sisis - who is due to meet his top aides in light of Friday's attack - has been struggling to contain an insurgency by Islamic militants.
In recent years, northern stretches of the Sinai have borne the brunt, however, attacks have increasingly targeted the Egyptian mainland in recent months.
Ten per cent of people in Egypt are Christians.
Click here to listen to Premier's Alex Williams speaking with His Grace, Bishop Angaelos: