The Republic of Ireland international started last season as first choice at the Stadium of Light, but a shoulder injury last November saw him usurped by Vito Mannone.
Sunderland have also added Costel Pantilimon to their goalkeeping options ahead of the new campaign, and Westwood was subsequently not offered a new deal by the Wearsiders.
The 29-year-old, who joined Sunderland from Coventry City in June 2011, has now signed a two-year deal with Wednesday.
Wednesday - who finished 16th in the Championship last term - were bought by Azerbaijani business Hafiz Mammadov last month.
Mammadov, who owns Ligue 1 outfit Lens, has yet to get ratification from the Football League for the purchase.
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