His tournament victories include the first place tie (+6−0=9) with Boris Spassky at the Chigorin Memorial in Sochi 1965, first at Maribor 1967 ahead of Samuel Reshevsky, first at Krems, and first at Amsterdam 1980 tied with Hans Ree. 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. At the Tel Aviv 1964 Chess Olympiad Unzicker scored 13.5 points playing first board for the West German team that won the bronze medal on the strength of a 3:1 team victory over the Soviet Union. Unzicker also shared fourth place (+2−1=14) with Lajos Portisch in the 1966 Piatigorsky Cup in Santa Monica, California. Only Boris Spassky, Bobby Fischer, and Bent Larsen finished ahead of Unzicker. 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. Unzicker finished second (+3−2=7) to Viktor Korchnoi at South Africa 1979. A retired judge, he was still playing chess as first board on the club team "Tarrasch Munich".