Seventeen out of eighteen 1977–78 Yugoslav First League clubs entered the competition at this stage (everyone except Osijek, who had been promoted to top level at the end of the 1976–77 season along with Kosovo-based minnows Trepča).
Borac Čačak were the only team from outside top level who managed to progress further after winning their away tie against Bosnian side Sloboda on penalties.
The last remaining member of the "Big Four" clubs, Dinamo Zagreb, were sensationally knocked out on penalties after they were held to a goalless draw by Trepča in Mitrovica.
In line with the rules adopted by the Football Association of Yugoslavia in the late 1960s the final was to be played as a one-legged tie in Belgrade in cases when both finalists hailed from outside the capital city.
The match was thus played at the Red Star Stadium on 24 May 1978, intentionally set so that it would coincide with Youth Day, a national holiday celebrated on 25 May and which doubled as the official commemoration of Josip Broz Tito's birthday, the cup's sponsor.
Reaching the 1978 cup final is still considered their biggest success in history and their only domestic silverware to date came when they won the Superleague of Kosovo in 2009–10.