In 1964, Burton portrayed Thomas Becket, the Archbishop of Canterbury who was martyred by Henry II of England, in the film adaptation of Jean Anouilh's historical play Becket. Both Alpert and historian Alex von Tunzelmann noted Burton gave an effective, restrained performance, contrasting with co-actor and friend Peter O'Toole's manic portrayal of Henry. Burton asked the film's director, Peter Glenville, not to oust him from the project like he had done for Adventure Story before accepting the role of Becket. Writing for The Christian Science Monitor, Peter Rainer labelled Burton as "extraordinary". Kenneth Turan of the Los Angeles Times appreciated Burton's on-screen chemistry with O'Toole and thought his portrayal of Becket served as "a reminder of how fine an actor Burton was". The film received twelve Oscar nominations, including Best Actor for both Burton and O'Toole; they lost to Harrison for My Fair Lady (1964). Burton and O'Toole also received nominations for Best Actor – Motion Picture Drama at the 22nd Golden Globe Awards, with O'Toole emerging victorious. Burton's triumph at the box office continued with his next appearance as the defrocked clergyman Dr. T. Lawrence Shannon in Tennessee Williams' The Night of the Iguana (1964) directed by John Huston; the film was also critically well received. Alpert believed Burton's success was due to how well he varied his acting with the three female characters, each of whom he tries to seduce differently: Ava Gardner (the randy hotel owner), Sue Lyon (the nubile American tourist), and Deborah Kerr (the poor, repressed artist). The success of Becket and The Night of the Iguana led Time magazine to term him "the new Mr. Box Office".
Burton helped Taylor make her stage debut in A Poetry Reading, a recitation of poems by the couple as well as anecdotes and quotes from the plays Burton had participated in thus far. The idea was conceived by Burton as a benefit performance for his mentor Philip, whose conservatory, the American Musical and Dramatic Academy, had fallen short of funds. A Poetry Reading opened at the Lunt-Fontanne on 21 June 1964 to a packed house; the couple received a standing ovation at the end of their performance. Burton remarked on Taylor's performance, "I didn't know she was going to be this good."
After Hamlet came to a close in August 1964, Burton and Taylor continued making films together. The first film after their marriage, The Sandpiper, was poorly received but still became a commercially successful venture. According to Bragg, the films they made during the mid-1960s contained a lot of innuendos that referred directly to their private lives. Burton went on to star opposite Claire Bloom and Oskar Werner in The Spy Who Came in from the Cold (1965), a Cold War espionage story about a British Intelligence agent, Alec Leamas (Burton), who is sent to East Germany on a mission to find and expose a mole working within his organisation for an East German Intelligence officer, Hans-Dieter Mundt (Peter van Eyck). Martin Ritt, the film's director and producer, wanted Burton's character to exhibit more anonymity, which meant no display of eloquent speeches or intense emotional moments. Bragg believed this decision worried Burton, as he had generated his reputation as an actor with those exact traits, and wondered how the film's would turn out. Ritt, a non-drinker, was displeased with Burton's drinking habits as he felt it "lacked a certain discipline" and expected the same level of commitment from him as everyone else during filming. In spite of their differences, Alpert notes that the film transpired well. Based on the 1963 novel of the same name by John le Carré, The Spy Who Came in from the Cold garnered positive reviews, with Fernando F. Croce of Slant Magazine describing Burton's performance as more of "tragic patsy than swashbuckler" and believed his scenes with Werner "have sharp doses of suspicion, cynicism and sadness". Dave Kehr of the Chicago Reader called the film "Grim, monotonous, and rather facile", he found Burton's role had "some honest poignancy". Variety thought Burton fitted "neatly into the role of the apparently burned out British agent". Burton also made a brief appearance the same year in Clive Donner's comedy What's New Pussycat? as a man who meets the womaniser Michael James (O'Toole) in a bar.