Mr Snowden presented his inaugural address via video-link to staff and students during the ceremony in the University of Glasgow's Bute Hall.
Chris Cassells, a student who nominated Snowden for the rectorship, said: I hope he will have the opportunity to stand up for Glasgow students."
Jessica McGrellis, president of the university's Student Representative Council (SRC) said the body will do everything it can to facilitate his involvement in student affairs.
She added: "I'm not sure how, but as long as there is good communication I am sure it will happen."
Another student at the inauguration ceremony said: "I think it is a good thing for the university. It's not going to show the world what to do, but it is a significant statement and with previous rectors being actors from EastEnders, it is a lot better than that."
Mr Snowden, speaking via video-link from an undisclosed location in Russia, was greeted by applause and cheering.
He said: "First off, I'd like to give my thanks to everyone at the university. To the student body, to the SRC, to everyone who participated in the election whether they were candidates or voters.
"I am disappointed and I must apologise for being unable to attend in person, but unfortunately I have discovered that I am barred from entering the UK on the grounds that my presence is considered detrimental to the public. I do think it is fair to say that the election shows the students of this university have a different opinion and I am honoured."
Mr Snowden went on to argue that, in a democracy, people have a right to know the policies of their government.
"We may not need the names and identities of every target of surveillance, of every active operation, but we should know the general outlines of what the government is doing in our name and particularly what the government is doing against us," he said.
Mr Snowden beat cyclist Graeme Obree, author Alan Bissett and the Rev Kelvin Holdsworth to become rector.