O'Conor focused his career primarily around Shakespearean roles, beginning with his 1939 stage debut, playing three bit parts in Julius Caesar, and extending to Polonius in Hamlet, a role he performed at the Old Vic in London, on Broadway, and on early television, in The DuPont Show of the Month. In later TV adaptations, he would appear as Friar Lawrence in Romeo and Juliet and Lodovico in Othello.
In film, O'Conor played the kindly Mr. Brownlow in the movie version of Oliver! and a coroner in the Hammer Studios horror film The Gorgon. The actor also appeared in several period films, including The Devil-Ship Pirates (again with Lee, as Don Jose Margella), Anne of the Thousand Days (as Bishop Fisher), Elizabeth (the Earl of Derby), and The Messenger: The Story of Joan of Arc (as an inquisitor). His television work included Old Jolyon in the 1967 serial The Forsyte Saga, as well as The Barchester Chronicles and The Glass (2001, his final role).