The Republic of Ireland international was available on a free transfer after he was released by Sunderland at the end of his contract last week.
Westwood joined Sunderland three years ago from Coventry City, but he made only 24 appearances during his time at the Stadium of Light.
The 29-year-old featured 11 times last season and had started the campaign as Sunderland’s first-choice keeper.
However, a shoulder injury kept Westwood sidelined from November before he was forced to have surgery on the problem in January.
Costel Pantilimon’s arrival on Wearside to provide competition for Vito Mannone left Westwood surplus to requirements and he has now finalised a move to Hillsborough.