Jacobsen was born in Copenhagen. He lived in France from 1947-69. From 1962-81, he was a professor at Kunstakademie der Bildenden Künste in Munich. In 1969 he moved to Tågelund, west of Egtved, Denmark. From 1976 to 1985 he was professor at the Royal Danish Academy of Art, Copenhagen. From 1986 to 1991 he worked with Jean Clareboudt to create a sculpture park at Tørskind Gravel Pit near Egtved and Vejle. He worked closely together with his friend and son in law, Bernard Leauté. Jacobsen also had a connection to Asger Jorn and the CoBrA artists, but he never was a member of their group. In 1974 he was awarded the Prince Eugen Medal by the King of Sweden.
He died in 1993, aged 80, at home in Tågelund. It was planned to raise a 60 meter high sculpture in Copenhagen in the year 2012, financed by Robert Jacobsen's foundation but the city council did not want the sculpture.