The 70-year-old, who had served as archbishop since 2012 and was one of the most senior figures in the Catholic Church in Scotland, died on January 13 after contracting Covid-19.
His funeral mass was held at St Andrew's Cathedral in Glasgow on Thursday.
The archbishop tested positive for coronavirus shortly after Christmas and had been self-isolating at home.
Bishop Hugh Gilbert, president of the Bishops' Conference of Scotland, said in his homily: "The image that comes to me is of a great tree felled unexpectedly in the middle of the night - storm Covid.
"And only when we woke up the day following did we begin to divine what had happened, did we begin to grasp the depths of its roots, to see the space this tree occupied, the shelter it gave, and what we've personally and collectively lost.
"This uprooting has changed the landscapes of so many lives.
"Looking at it from our side, we are commending to God today someone who wasn't small in any sense, someone of gravitas, and someone in whom head and heart came together, possessed of intellectual force and clarity and at the same time of great human warmth."
He said the archbishop cherished pastoral assignments and had a gift for friendship and insight into people.
He said: "As a pastor, especially here in Glasgowand for a while in Edinburgh too, he had plenty of valleys of darkness to walk through, with others, with unsettled priests, survivors of abuse, victims of accidents, and he did so in such a genuine, heartfelt way.
"The bin lorry episode, the helicopter on the roof, his concern for asylum seekers."
Under current coronavirus regulations, only 20 of the archbishop's closest family and friends were able to attend the service.
The Pope has expressed his sorrow at the death in a message received from Cardinal Parolin, his secretary of state.
The message reads: "His Holiness Pope Francis was deeply saddened to learn of the untimely death of Archbishop Philip Tartaglia, and he offers heartfelt condolence and the assurance of his spiritual closeness to the clergy, religious and lay faithful of the Archdiocese of Glasgow.
"With gratitude for the late archbishop's years of steadfast service to the building up of the Church in Scotland in obedience to the Gospel, in dedication to the works of the apostolate and in fidelity to the See of Peter, His Holiness willingly commends the soul of this much-loved pastor to the boundless mercy of Christ the Good Shepherd."
Archbishop Tartaglia was born in Glasgow on January 11 1951.
He was ordained a priest by Archbishop Thomas Winning in the Church of Our Lady of Good Counsel, Dennistoun, on June 30 1975.
Pope Benedict XVI nominated him Bishop of Paisley and he was ordained on November 202005.
He was appointed Archbishop of Glasgow on July 24 2012 and installed at St Andrew's Cathedral on September 8 of that year.
He died on January 13, which is the Feast of St Mungo - the patron saint of Glasgow.