Ivkov won numerous top-class events during his career; notable tournament triumphs include Mar del Plata 1955, Buenos Aires 1955, Beverwijk 1961, Zagreb 1965, Sarajevo 1967, Amsterdam-IBM 1974, and Moscow 1999.

Ivkov earned his National Master title in 1949 at age 16, by placing shared 4th–7th in the Yugoslav Championship at Zagreb, with 11/19; the winner was Svetozar Gligorić.

Ivkov earned his first international event opportunity at Bled 1950, sharing 5th–6th places with 7½/14; the winner was Miguel Najdorf.

In this tournament, which featured some of the world's best players, Ivkov defeated well-known stars such as Hermann Pilnik, Milan Vidmar Sr., and Vasja Pirc.

After this impressive debut, he was selected to represent Yugoslavia in team matches against the United States and the Netherlands in 1950 and against West Germany in 1951.

Ivkov won the inaugural World Junior Chess Championship in 1951 at Birmingham; this tournament was established for players under age 20.

Ivkov broke through into the group of the world's elite players with two brilliant results in strong tournaments in Argentina.

He won at Mar del Plata 1955 with 11½/15, ahead of Najdorf, Gligoric, Szabo, Luděk Pachman, Panno and Pilnik.

Then he won again at Buenos Aires 1955 with 13/17, ahead of Gligoric, Pilnik, László Szabó, Bisguier, Luděk Pachman, Rosseto, Panno and Donner.

In the Yugoslav Championship at Novi Sad 1955, Ivkov had his best national result to date with a shared 3rd–4th place on 10½/17; the winner was Karaklajic.

[7] Ivkov made his first appearance at the European team level in the inaugural event at Vienna 1957, and was chosen on five further occasions.

Ivkov also participated in 16 editions of the USSR versus Yugoslavia matches from 1956 to 1979, more than any other Yugoslav or Soviet player.

[8] Ivkov narrowly missed becoming Yugoslav champion at Sombor 1957 when he scored 14/22 to share 2nd–4th places behind Gligoric.

Then at Zagreb 1965, Ivkov scored perhaps his most impressive career victory, sharing the title with Wolfgang Uhlmann on 13½/19, ahead of World Champion Tigran Petrosian, Lajos Portisch, Larsen, and Bronstein.

He had a disappointing result at the elite Piatigorsky Cup tournament at Santa Monica, California, finishing with a minus score, as Spassky won.

From the Raach Zonal 1969, he advanced to the Interzonal after sharing 2nd–5th places with Portisch, Jan Smejkal, and Ulf Andersson, then winning a further playoff.

Ivkov shared the title at the 1972 Caorle Zonal tournament on 12/17 with Ljubomir Ljubojević, to qualify for the next year's Interzonal.

Ivkov shared 2nd–9th places in the 1998 World Senior Championship at Grieskirchen on 8/11 behind champion Vladimir Bagirov.

His most impressive late-career feat was a shared 1st–2nd place in the strong Tigran Petrosian Memorial at Moscow 1999 with 5/9, along with Portisch.

Ivkov tied 2nd–4th places in the 25th anniversary Paul Keres Memorial at Vancouver with 7½/10, behind winner Vladimir Epishin.

[10] Ivkov defeated five World Champions: Vasily Smyslov, Mikhail Tal, Tigran Petrosian, Bobby Fischer, and Anatoly Karpov.