A midfielder, Götz began his career in East Germany with Vorwärts and later BFC Dynamo. In 1983, before a European Cup match against Partizan Belgrade, he escaped and fled to the west. For this rule breach, he was banned by FIFA for one year, but was able to stay in the west, joining Bayer Leverkusen, where he stayed for five years, moving to 1. FC Köln in 1988. With Bayer Leverkusen, he won the UEFA Cup in 1988. He scored in the second leg of the final against Espanyol, one of three goals needed to equal a 3–0 deficit. Leverkusen eventually went on to win the game on penalties. He had spells with Galatasaray (1992–1994), Saarbrücken (1994–1995), and Hertha BSC (1995–1997) before retiring, to take up the role of Hertha's reserve team manager.
He was briefly Hertha's caretaker manager during 2002, and was re-appointed as full-time manager in 2004, having managed 1860 München the previous season. Götz was sacked by Hertha on 10 April 2007. On 15 December 2008, he took over as manager of Holstein Kiel and was dismissed on 17 September 2009.
In 2011 Götz was appointed as manager of the Vietnam national football team. After some positive games in national-level team, however their under-23 team disappointed, and he was fired on 23 December 2011, just six months after taking charge.