The 24-year-old arrives at the Stadium of Light just five days prior to the start of the Premier League campaign - becoming Paolo Di Canio's 10th summer acquisition.
Celustka began his career at FC Tescoma Zlín in 2007 before leaving in 2009 for Slavia Prague, where he made 41 league appearances.
Following a short spell at Palermo, Celustka went on to rack up 55 appearances for Turkish Super Lig club Trabzonspor. He also won 18 Czech Republic Under-21 caps.
The Black Cats kick off their domestic season against Fulham at home on Saturday.