And she praised the city for ensuring local people can use the £113million sports complex in the run up to the Commonwealth Games.
Dame Tessa, who served as Minister for the Olympics and Secretary of State for Culture Media and Sport in the run-up to the London 2012 Games resigned from her role as Shadow Olympics Minister last month.
She was given a tour of the East End venue by the city council's deputy leader and Commonwealth Games spokesman Archie Graham.
Dame Tessa said: "This is a fantastic facility. It is great that the local community will be using it on a regular basis. That is unique and something Glasgow should be proud of."
More than 10,000 people visited the facility, which includes the Sir Chris Hoy Velodrome, when it opened to the public last weekend.
Mr Graham, who is also chairman of Glasgow Life, said: "The Emirates Arena was built for the people of Glasgow and they have already been coming through the doors in their thousands. We want them to be inspired."
The arena can accommodate up to 6500 spectators and stage sports including athletics, badminton, boxing, gymnastics, judo and netball.