The actor René Auberjonois, who has died aged 79 of cancer, had a distinctive face – beaky, birdlike, inquisitive – and a gift for comic haughtiness. His biggest successes came on television: as well as appearing in guest roles on everything from The Rockford Files to Grey’s Anatomy, Starsky and Hutch to Frasier, he was a regular on three popular and vastly different shows, each of which required him to play variations on his stuffed-shirt persona.

In the sitcom Benson, a spin-off from the irreverent 1970s television hit Soap, he was the prissy, hypochondriac chief of staff to a US governor; he joined the show in its second season in 1980 and remained on board until its cancellation six years later. He was seen throughout the seven seasons of Star Trek: Deep Space Nine (1993-99) as Odo, a pebble-faced shapeshifter who could change into anything from a seagull to a spinning top, but whose natural state was a gelatinous mass in a bucket. He was back in his own skin for all five seasons of Boston Legal (2004-08), in which he played a by-the-book partner in a law firm whose daughter is a drug addict.

Admirers of 70s US cinema knew him from his work with Robert Altman, who cast him in four consecutive films. Auberjonois had been in rehearsals for the Broadway production of Fire! when Altman came calling. “I hope it flops,” the director said of the play. And it did.

Had it not, the actor would have missed out on the part that launched his screen career. As the chaplain Father Mulcahy, also known as “Dago Red”, in Altman’s 1970 Korean war comedy, M*A*S*H, he was effectively the straight man, a sober fussbudget among the anarchy of the rest of the film. Auberjonois turned down the offer to reprise his role in the TV series.

Always alert to an eccentric presence, Altman cast him in his next three films. In Brewster McCloud (1970) he was an ornithologist who gradually turns into a bird, feather by feather, over the course of the picture.

In the director’s masterpiece, McCabe & Mrs Miller (1971), he was a ferrety, sycophantic innkeeper currying favour with the big guns in town and ready to switch sides when the odds change. And in the psychological thriller Images (1972) he played the husband of a writer (Susannah York) undergoing a terrifying breakdown. He was also one of many actors later called upon by Altman to add background colour by appearing as themselves in the Hollywood satire The Player (1992).

Auberjonois, who shared his name with his grandfather, a Swiss post-impressionist painter, was born in New York and spent some of his early childhood in London and Paris. His father, Fernand, was a foreign correspondent for the US press who had enjoyed a wildly varied career: he had once been Katharine Hepburn’s private French tutor, as well as an aide to generals Patton and Eisenhower. He married Laure (nee Murat), who had two daughters from a previous marriage.

When the family returned from Paris to New York, they lived near the actor and producer John Houseman, who became a mentor to Auberjonois and secured him, at the age of 16, an apprenticeship with the American Shakespeare Theatre company in Stratford, Connecticut, where he was artistic director. Auberjonois was educated at the Carnegie Institute of Technology.

He began his stage career with the American Conservatory Theater company in Pittsburgh and San Francisco, and appeared on Broadway in 1968 as the fool opposite Lee J Cobb (who personally approved him for the role) in King Lear. When hHe won a Tony award in 1970 for playing a flamboyant designer in Coco, a musical about Coco Chanel, he was less than thrilled. “I wasn’t able to enjoy the moment, or be grateful for it because I somehow wanted it to be for something more than playing a fop. It may be my Swiss Protestant ethic for hard work, but it was confusing to be rewarded for something easy.”

He received three further Tony nominations: for The Good Doctor, Neil Simon’s 1973 comic take on Chekhov, in which he starred with Christopher Plummer; Big River: The Adventures of Huckleberry Finn, the 1985 musical with Auberjonois (looking, one critic noted, “like a Dr Seuss cartoon”) as a con artist; and the 1989 comedy-musical City of Angels, set in 40s Hollywood, where his performance as the egomaniacal and corruptible producer Buddy Fidler was described by the New York Times as “gleefully smarmy”.

His screen credits exceeded 200, and included the disaster movie The Hindenburg (1975) and a well-liked spoof of that genre The Big Bus (1976). He was in the thriller Eyes of Laura Mars (1978), the comedy Where the Buffalo Roam (1980) and the American war of independence drama The Patriot (2000). He also had innumerable voiceover roles in animation – the best known of which was the chef who sings Les Poissons in Disney’s The Little Mermaid (1989) – and in videogames.

Asked to cite his most demanding part, he singled out playing a paediatric heart surgeon with Tourette syndrome on a 1999 episode of Chicago Hope.

“A year later at some Beverly Hills event to raise consciousness about Tourette’s, they showed a clip of it,” he said. Recently, he was excellent as an intractable curmudgeon in an emotionally tense scene with Michelle Williams in Certain Women (2016), a portmanteau drama by Kelly Reichardt with whom he worked again on First Cow (2019), her film about 19th-century fur trappers.

Asked in 1985 how he thought he came across on camera, he suggested “snobbish”, “effete” and “aristocratic” before admitting: “I’ve grown to accept it.”

He is survived by his wife, Judith (nee Mihalyi), whom he married in 1963, their children, Tessa and Remy, three grandchildren and his two half-sisters.

• René Marie Murat Auberjonois, actor, born 1 June 1940; died 8 December 2019

• This article was amended on 11 December 2019. The Patriot is a drama about the American war of independence, not the civil war.