Unzicker began to play tournaments abroad in 1948 as Germany was struggling to rebuild after the war, and achieved the grandmaster title in 1954.

In 1950, Unzicker shared the prize for best top-board score (+9−1=4) with Miguel Najdorf for his performance on first board for the West German team at the Dubrovnik Chess Olympiad.

Unzicker placed ahead of world champion Tigran Petrosian, Samuel Reshevsky, Miguel Najdorf, Borislav Ivkov, and Hein Donner.

At Hastings 1969–70, Unzicker finished second (+4−0=5) after Lajos Portisch and ahead of Svetozar Gligorić and former world champion Vasily Smyslov.

In 2005, Unzicker celebrated his 80th birthday with his wife Freia, his three sons and their wives and three grandchildren, and a tournament with Karpov, Korchnoi and Spassky took place in his honor.