It’s been close to a quarter of a century since I’ve read any of them, so I can’t think of a suitable one off the top of my head, but surely one of John Irving’s novels must have just what you’re looking for.
Paging Hari Seldon.
Hari Seldon, please pick up the White Courtesy Phone.
Robert B. Parker’s Spencer, Jesse Stone, and Sunny Randall novels all share a common world which includes Susan Silverman, a psychiatrist. She is Spenser’s significant other and Sunny Randall’s therapist. I can’t recall if Jesse Stone has had any direct interactions or not with Susan.
-DF
“I now call Doctor McCoy to the stand.”
“Service rank: Lieutenant Commander. Position: ship’s surgeon. Current assignment: USS Enterprise. Commendations: Legion of Honour. Awards of Valour. Decorated by Starfleet surgeons.”
“Doctor, you are, on the record, an expert in psychology, especially space psychology. Patterns which develop in the close quarters of a ship during long voyages in deep space.”
- from COURT-MARTIAL.
“I’m a surgeon, not a psychiatrist!”
The City on the Edge of Forever
“Yes, Mister Spock, what is it?”
“I picked this up from Doctor McCoy’s log. We have a crewmember aboard who’s showing signs of stress and fatigue. Reaction time down nine to twelve percent, associational reading norm minus three.”
“That’s much too low a rating.”
“He’s becoming irritable and quarrelsome, yet he refuses to take rest and rehabilitation. Now, he has that right, but we’ve found…”
“A crewman’s right ends where the safety of the ship begins. Now, that man will go ashore on my orders. What’s his name?”
“James Kirk. Enjoy yourself, Captain; it’s an interesting planet. You’ll find it quite pleasant – very much like your Earth.”
- Shore Leave
So, sure, he’s not a psychiatrist; he’s just a trained medical professional who evaluates, diagnoses, treats, and studies behavior and mental processes.
A single line, the final one in the book.
Dr. Luther Brooks in No Way Out, but that’s a major role.
Dr. Patel in Wally Lamb’s “I Know This Much is True” would fit what you’re looking for. She’s an Indian psychologist that helps the main character deal with having a mentally ill brother.
Constance Leidl in Kate Wilhelm’s Constance and Charlie mysteries. She is warm, empathic, and her use of Ericksonian hypnotherapy is rather good. Even though the mysteries concentrate on, well, mysteries rather than her therapeutic practice, you can see her abilities. Start with The Hamlet Trap.
Dr. Stanley Keyworth, the psychiatrist played by Adam Arkin on* the West Wing*. Suffice to say that he is to other psychiatrists what the regular cast of the show is to actual politicians - smarter, wittier, more idealistic and more professional.
In SPHERE, Michael Crichton makes the protagonist a psychologist – but he’s so low-key that he acts like a minor character, figuring that his teammates are the ones who should handle the exploration and even the hypothesizing about the problem du jour; they can analyze that, Norman’s job is analyzing them. He has no interest in playing explosives expert or scuba diver; he just wants those dynamic folks to – cooperate and get along with each other.
A few more:
Dr. Theodore Kriezler, in Caleb Carr’s The Alienist. Definitely an admirable character.
Max Lieberman a A Death in Vienna, by Frank Tallis
Freud himself in The White Hotel, by D.M. Thomas. A fictionalized version, so I think it counts.
Sorry…didn’t notice the minor character preference of the OP. Will rethink.
There’s also the page on “The Shrink” at TVTropes, which breaks fictional psychiatrists into three basic types: The Harmful Shrink; The Well-Meaning, But Dopey and Ineffective Shrink; and The Awesome Shrink. Presumably this third type is what the OP is looking for.
While they don’t involve psychiatrists as such, some of the novels in Susan Howatch’s “Church of England” series involve key scenes of what look very much like psychotherapy or psychoanalysis in an ecclesiastical context. For example, in the first novel of the series, Glittering Images, the main character has a nervous breakdown or psychological/emotional/spiritual crisis and spends time talking with Jon Darrow, a “spiritual director,” to get to the bottom of his issues. Darrow is a minor, and admirable, character in this book, but is the protagonist of the next book, in which he has his own crisis and undergoes his own spiritual therapy.
Thanks, all. Some great suggestions.
… never occurred to me that Wiki would have such a list. Sigh. The list is, of course, woefully incomplete since it doesn’t include the shrink from PSYCHO, nor Dr Chumley from the play HARVEY. 
Sounds a little like Captain Newman, M.D. by Leo Rosten. A good psychotherapist, but obviously not a minor character. Also loosely based on a real person.
Dr. Susan Calvin is a brilliant psychologist who shows up in Isaac Asimov’s I, Robot stories. Granted, she’s a robot psychologist. Details, details.
Freud also appears in the non-canonical Sherlock Holmes novel The Seven-Per-Cent Solution, although he’s a major character.
No mention of Dr. Melfi from The Sopranos? You could also mention her psychiatrist, played by Peter Bogdanovich.
There’s a minor character in Tom Clancy’s Without Remorse, who’s a psychiatrist and consults with the Baltimore police detective (Jack Ryan’s father) investigating a string of drug-related murders. Dr. Sydney Farber and he appears in maybe two scenes, IIRC.
Based on the crime scene evidence and methodology, he suggests the killer is some kind of commando/special forces type, later on even mentioning the SEALs, which were supposedly not well known at the time of the novel’s setting. A “man on a mission” who’s seriously pissed off and taking down drug dealers with a lot of skill and in all sorts of imaginative ways.
Not literature, but the regular Law & Order shrinks (Skoda, Olivet, & Huang) are all portrayed as competent.