Robert Eigenberger studied art history at the German University in Prague and wrote his doctoral thesis in 1913 on the sculptor Adam Krafft. The same year he became an intern and later an assistant at the k. k. Zentralkommission für Denkmalpflege (Imperial Royal Central Commission for Monument Conservation) in Vienna. During the First World War, he volunteered in May 1915 but was discharged six months later, having been seriously ill with typhoid fever, and was then employed collecting metal for the war effort. In 1917 he was appointed curator of the Paintings Gallery of the k. k. Akademie der bildenden Künste in Wien (Imperial Royal Academy of Fine Arts in Vienna). In his function from 1922 as director of the Paintings Gallery he reorganized the display and published a comprehensive inventory in 1927. He was also a freelance painter, winning the Österreichischer Staatspreis für Malerei (Austrian State Prize for Painting) in 1930 and was a member of the Vereinigung bildender Künstler Wiener Secession (Society of Fine Artists – Vienna Secession). In 1926 he was awarded the title of professor and from the summer semester 1927 was also an honorary lecturer in art history at the Viennese Akademie. From 1930 at the latest he was also an assessor and expert at the Dorotheum. In 1933 he was put in charge of restoration in the Paintings Gallery and gave a restoration course at the Akademie. He was subsequently appointed extraordinary professor and became head of the Department of Conservation and Technology in 1934. By his own admission, he carried out intelligence work for the illegal NSDAP from around 1934 but did not officially become a member until May 1938. At the same time, he changed his religion from Roman Catholic to "gottgläubig" ("religious"), became a supporting member of the SS and NS-Dozentenführer (NS senior lecturer) at the Viennese Akademie. In 1940 he was appointed ordinary professor and was given the Erinnerungsmedaille (Memorial Medal) on 13 March 1938 for his "National Socialist activity during its illegality". This was followed in 1942 by the Silbernes Treudienstehrenzeichen (Silver Loyalty Medal of Honour). Eigenberger was the Akademie's air raid officer responsible for salvaging works from the Paintings Gallery and deciding which items in the collection were to be stored for protection during the Second World War. From 1939, for example, he organized the transfer of over 600 objects to external depots inside and outside Vienna. Around 1,300 works, categorized by him as third-class, were left in the Academy building, around 520 of which were destroyed or disappeared when it was hit by a bomb on 12 March 1945. Moreover, 120 items stored in external depots were lost through looting and other causes.
Eigenberger was relieved of his position in June 1945 but was able to resume teaching at the Akademie in November 1946. According to a decision by the Federal President of 13 September 1947 he was classified as a lesser offender on account of his "exceptional knowledge", and in 1949 he was made extraordinary professor. That year the Federal Ministry of Education investigated Eigenberger, accusing him of having caused sustained damage during the restoration work carried out by him in 1938/1939. The expert opinions by Ludwig Baldass, Otto Benesch, Karl Garzarolli-Thurnlackh, Josef Hajsinek and Franz Sochor in early 1949 questioned Eigenberger's qualification as a restorer and head of the masterclass for restoration and technology. Eigenberger had no training as a restorer but was self-taught. As the committee established to examine the reproaches—four of the five members were professors at the Akademie—did not reach any satisfactory conclusions, the Ministry of Education called in an external expert in 1950/51, which led to protests by the teaching staff at the Akademie. The records do not show what consequences this had for Eigenberger. At the staff meeting of 1 June 1951 he was elected rector of the Akademie for the academic years 1951/52 and 1952/53, and he held this position again in 1954/55. He was pro-rector in 1953/54 and 1955–1957 and became an ordinary professor in 1955. After his retirement on 27 July 1961 he remained head of the masterclass for conservation and technology and also gave a course in this subject. From 1962 to 1965 he was chairman of the Akademie's Institut für sakrale Kunst (Institute for Sacral Art). He retired from active service on 30 September 1965. After receiving the Goldenes Lorbeer (Golden Laurel) of the Vienna Künstlerhaus in 1960, he was awarded the Große Ehrenurkunde (Great Certificate of Honour) of the Academy of Fine Arts in Vienna in 1978 and honorary membership of the Genossenschaft bildender Künstler Wiens (Künstlerhaus) in 1977.