Philip Tartaglia, 61, who is the existing Bishop of Paisley, was appointed by Pope Benedict XVI today.
He will take over from Archbishop Mario Conti, who is retiring after 10 years in the post.
He has recently come to the fore over the same-sex marriage debate, clashing with the Scottish Government over its proposals to allow gay couples to marry in religious ceremonies.
He previously sent a strongly worded submission to the First Minister and said governments "do not have the authority to say what marriage is or to change its nature or to decree that people of the same sex can marry".
The new Archbishop-Elect, who will take up the post on September 8, said: "I am conscious of the historic place of the archdiocese of Glasgow in the history of Christianity in Scotland and of its importance for the Catholic community in particular. It is a great honour for me to be appointed Archbishop of my native city and diocese."
Bishop Tartaglia said he had been "very happy and fulfilled" as Bishop of Paisley, where he has served since late 2005.
"I have loved my diocese," he said. "The priests and people of the diocese of Paisley will always be in my heart. And I would want to serve the archdiocese of Glasgow with the same affection and devotion. I know I can only be a good bishop with help from above, so I ask everyone to pray for me."
Archbishop Conti said: "I am delighted that the Holy Father has appointed Bishop Philip as my successor. I have known him for more than 30 years and I have the greatest admiration for his gifts of leadership, intelligence and pastoral sensitivity. The archdiocese will be in very good hands. I hope to remain of service to the people of the archdiocese and I will place myself at the service of my successor to assist him in whatever way he may wish in the life of the Church in Glasgow."
Scotland's most senior Catholic, Cardinal Keith O'Brien, paid tribute to the outgoing Archbishop.
"He will indeed be missed in his own archdiocese, and beyond," he said. "However, alert as always, I am sure that he will be available to add to any discussion about ways forward for the Catholic Church in Scotland and will continue to show himself as an ardent pastor in our country. The successor appointed to Archbishop Conti - Bishop Philip Tartaglia, Bishop of Paisley - has greatly endeared himself to the priests and people of his own diocese and I am sure that while they deeply regret his move from Paisley, they will also be happy to know that he will be able to continue his heavy responsibilities in the larger archdiocese."
The archdiocese of Glasgow is the largest of Scotland's eight dioceses with an estimated Catholic population of 200,000.
It comprises 95 parishes and is served by 203 priests.
Gordon Matheson, leader of Glasgow City Council, said: "I warmly welcome the appointment by Pope Benedict XVI of Bishop Philip Tartaglia as the new Archbishop of Glasgow. I look forward to working closely with him in the years ahead. I know that he will be constantly supported by the Catholic community of Glasgow as he fulfils his heavy responsibilities, and will benefit too from the co-operation of all people of goodwill in the city. I would also like to pay tribute to Archbishop Mario Conti who has been an exemplar pastor and churchman who has served this city with great distinction. I wish him every health and happiness in retirement and success in whatever endeavours he chooses to undertake. I am confident that he will continue to play a full role in the life of the city."