The £16m pedestrian-only link is the first new crossing to span the Thames in central London since the opening of Tower Bridge in 1894.
But work on the project has fallen behind schedule, postponing its opening until 10 June.
When open, the 320m-long crossing will run from St Paul's Cathedral north of the river to the new Tate Modern gallery and the Globe Theatre in Southwark.
Specially composed fanfares sounded from each side of the river as the Queen walked past brightly coloured flags and onto the bridge.
She was accompanied by an entourage including Prince Philip, the Lord Mayor of London and the Bishop of London.
The bishop gave a short prayer after which the Queen dedicated the bridge.
She said: "I dedicate this bridge as a symbol of the new millennium to the people of Southwark and of the City of London and to all who shall pass over it from all over the world."
The band of the Goldsmiths' School of Music struck up on the south side of the river and fireworks echoed across the Thames.
Before the dedication ceremony the Queen attended a service at St Paul's Cathedral with the Duke of Edinburgh and new London mayor Ken Livingstone.
Mr Livingstone said the bridge will "make a tremendous difference to the South Bank with all the development going on there".
He added: "It will be so good to actually walk across the river peacefully, without cars and trains thundering by."
Designed by architect Sir Norman Foster, the bridge will be lit at night to form a "blade of light" across the river.
More than half the money for the bridge has come from Lottery funds. The Millennium Commission has contributed more than £7.1m and a further £3.5m has come from the Corporation of London.
However the bridge is reported to have exceeded its budget by more than £2m.
It was originally hoped the bridge would be open this week to coincide with the opening of the new Tate gallery.
The decision to delay the public opening until June was made late last year.