Daučík played as a defender for 1. ČsŠK Bratislava, Slavia Prague and the Czechoslovakia national team. He was only the third Slovak player ever to play for Czechoslovakia. He represented them at both the 1934 and 1938 FIFA World Cups. He also coached Slovan Bratislava between 1942 and 1946 and again in 1948. In 1948, he coached Czechoslovakia for two matches.
Daučík arrived in Spain as coach of Hungaria, a team made up of refugees fleeing the various Communist regimes of Eastern Europe. The team included his brother-in-law, Ladislao Kubala, who had married Daučík's sister, Anna Viola Daučík, in 1947. Hungaria played a series of friendlies against Madrid Select XI, a Spain XI and Español and during these matches, Kubala was spotted by Josep Samitier, then chief scout at Barcelona. Kubala was offered a contract and as part of the deal, Daučík became Barcelona coach in 1950.
Daučík coached Barcelona during one of the club's most successful periods. As well as Nicolae Simatoc–Kubala, the team also included Juan Zambudio Velasco, Antoni Ramallets and Joan Segarra. During his four seasons in charge, the club won two consecutive La Liga/Copa del Generalísimo doubles in 1952 and 1953. In 1952, Barcelona won five different trophies, including La Liga, the Copa del Generalísimo, the Copa Latino and the Copa Eva Duarte. His contract ended in 1954 and after clashing with some players, he left the club.
Daučík subsequently joined Atlético Bilbao, where he won the Copa del Generalísimo in 1955 and his third Liga/Copa del Generalísimo double in 1956. Then with a squad that included Agustín Gaínza, Armando Merodio and Jesús Garay, Daučík then led Atlético to the quarter-finals of the European Cup in 1957. En route to the final, they eliminated Porto and Honvéd before losing 6–5 on aggregate to Manchester United.
The 1957–58 season saw Daučík take charge of Atlético Madrid and lead them to second place in La Liga. This resulted in Atlético qualifying for the 1958–59 seasons European Cup since the winners, Real Madrid, were also the reigning European champions. Inspired by Vavá and Enrique Collar, Atlético reached the semi-finals after beating Drumcondra FC, CSKA Sofia and Schalke 04. In the semi-finals, they met Real Madrid, who won the first leg 2–1 at the Santiago Bernabéu Stadium while Atlético won 1–0 at the Metropolitano. If away goals had counted double, Atlético would have progressed to the final. However, the tie went to a replay and Real won 2–1 in Zaragoza.