Webster received his first piano lessons from his father, who was the founder and director of the Conservatory of Pittsburgh. In 1921 the family moved to Paris. Here Webster studied at the Conservatoire de Paris with Isidore Philipp and was the first (and only) American who received the first prize of the Conservatory in piano.
He was also a student of Nadia Boulanger at Fontainebleau and Robert Casadesus at the American Conservatory. Here he received a first prize in 1922. In the early 1930s he studied with Artur Schnabel in Berlin.
Since the 1920s, Webster gave concerts in Paris throughout Europe. He was friends with Maurice Ravel and Jacques Durand, the editor of Debussy and played in 1924 at sixteen with the violinist Samuel Dushkin the premiere of Ravel's Tzigane. In the Paris period he also began to teach; among his pupils were Hephzibah Menuhin and Yaltah.
1934 Webster debuted in the U.S. with the New York Philharmonic Orchestra in the Second Piano Concerto by Edward MacDowell. A few weeks later he gave his first piano recital at Carnegie Hall in New York City. He joined in the following years with the most prestigious orchestras in the United States on as well as a piano soloist and chamber musician. His repertoire included classical piano literature from Bach to Beethoven, to the avant-garde of the 20th century. In particular, he was regarded as a specialist in the works of Ravel and Debussy.
He played premieres and often premiere recordings of works by American composers such as Elliott Carter, Roger Sessions, William Schuman, Louise Talma, Robert Helps and David Diamond and also played premiere recordings of compositions by Alban Berg, Arnold Schoenberg and Igor Stravinsky one. To mark the 50th anniversary of the death of Debussy, he played his entire piano works in a three-part concert series, among others in New York and Chicago.
From 1940 to 1946 taught Webster at New England Conservatory, then until retirement in 1990 at the Juilliard School of Music. He also worked as a visiting professor at numerous conservatories and universities in the U.S. and was from 1961 to 1970 performing artist at the Aspen Music Festival. Among his pupils were, among others, Paul Jacobs and Michel Block, Thomas Schumacher, Thomas Mastroianni and Raymond Jackson.
Webster was married since 1940 with singer Frances Brown, with whom he also appeared together. His son Michael Webster was known as a clarinetist, conductor and composer.