Scottish Premiership rivals Partick Thistle, Championship outfit Hibernian and clubs from England were among those interested in his signature.
Hateley, who previously had a spell in Scotland with Motherwell, has signed a two-year contract at Dens Park.
He made 29 appearances last season as Slask finished 10th in the table.
Born in Monaco while his father, Mark, was playing there in the French top flight, Hateley started his own football career with Reading.
After a loan spell with Basingstoke Town, he moved to Motherwell in 2009 and went on to make 150 appearances for the Scottish top-flight outfit.
Having rejected a new contract at Fir Park, Hateley had a four-month spell with Tranmere Rovers before moving to Poland.
Dundee said on their website: "He has been much sought after over the past few weeks, with a number of clubs vying for his signature."
Although Hateley can play in midfield, he can also play right-back and Dundee manager Paul Hartley has been seeking a replacement in that position after Paul McGinn moved to Chesterfield.
Ziggy Gordon, who rejected a new contract with Hamilton Academical, had chosen to join Partick Thistle instead of Dundee earlier this summer.
Dundee have won the race to sign Tom Hateley after the 26-year-old right-back/midfielder's release by Polish top-flight club Slask Wroclaw.