Father Nicholas, the senior parish priest, described to Mr Putin the history of the cathedral, which is frequented by students.
"The iconostasis is very beautiful. Downright majestic," the Prime Minister remarked, to which Metropolitan Barnabas of Cheboksary and Chuvashia replied: "It is in the old style. Young people need both old-style and modern things."
"The Church and the Government will work hand-in-hand to revitalise Russia so that it can prosper," he went on.
"So we will," Mr Putin agreed.
The Prime Minister gave a 19th century icon of St John the Baptist to the cathedral, and Father Nicholas presented him with an icon by local painters.
Despite the intense cold, people gathered to meet Mr Putin near the university and cathedral. He spoke with them after leaving the cathedral.
St Tatiana's Cathedral of the Intercession is close to several colleges and so it is visited by many students. The cathedral contains the relics of St Innocent the Bishop of Irkutsk and a venerated icon of St Tatiana that includes her relics. Both items were transferred there from the Cathedral of the Presentation of the Virgin Mary on September 16, 2006.