Once he crossed the line ahead of Germany's Bert Grabsch, bronze looked assured and the question of silver came down to the stamina of a struggling Cancellara.
Olympic time trial champion Cancellara, who won each of the four World Championships trials he entered between 2006 and 2010, saw his advantage over Wiggins crumble in patchy, light rain around the Danish circuit.
He lost his grip coming into a corner near the home straight and ended up narrowly behind the Briton.
Wiggins, who has won three Olympic and six world titles on the track, added: "After all the track medals and Olympic medals, it's nice to come to the Road Worlds and get a medal.
"To be up there with those two guys completes my transition to a road rider from track."
Wiggins and Millar will next appear in Britain's powerful seven-man lead-out train for Mark Cavendish in Sunday's road race.
Sprinter Cavendish believes he will never have a better opportunity to become the world champion than on the flat Copenhagen course - which invites a bunch sprint finish - helped by a team others have hailed as
the strongest British line-up
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