Google Books finds instances of the phrase "the smartest guy/person in the room" from long before it became a popular saying in mainstream U.S. society. Here are six from 1983 or earlier. From a 28-page pamphlet titled "Detroit. Twelfth Street in Perspective" (1967) [snippet view]:
If anyone can salvage a political situation, it is [Jerome] Cavanagh. When men of concern gather in his office, Cavanagh is usually the smartest guy in the room. He is blessed with mental toughness, a sense of humor, and an Irish fatalism which holds that life was never supposed to be that good anyway. At this point in his life, he is like the exciting, undefeated boxer who comes out of the ring from his first great beating.
Cavanagh was mayor of Detroit from 1962 to 1970. This pamphlet appears to have been issued in the wake of the 1967 Detroit riot.
From Ralph Regula, "Class of 1973—America Needs You," in Congressional Record (June 27, 1973):
No person achieves greatness unless he has maturity in these four essential dimensions of life. Each of us, as we seek to grow in understanding, recognizes that there must be balance in life. It is not enough to be the smartest person in the room, nor is it enough to be "in" with the socially elite. Like a sturdy table, a person must have four underpinnings long enough to reach the ground. dimensions.
From Biography News (March/April 1975) [combined snippets]:
"You want to know about Bob Nix?" asks an aide to another Pennsylvania congressman. "Let me start with the good stuff. In meetings, he is incisive, sometimes the smartest person in the room. He'll get right to the heart of something after everyone has been running at the mouth and he'll be able to say what has to be done. If he doesn't know anything about the subject, he says nothing rather than say something dumb. That's the good stuff. The bad stuff is that I've never seen him take leadership on anything. You go into his office and it always seems empty. Nobody seems to be doing anything."
From testimony by Ms. Lynch in Oversight Hearing on the Older American Nutrition Programs: Hearing Before the Subcommittee on Human Resources of the Committee on Education and Labor, House of Representatives, Ninety-sixth Congress, First Session (April 3, 1979) [combined snippets]:
The State nutritionist told me about this and she said that she did tell the area agency director, in fact, that she had to because of the grandfather clause in the Older Americans Act in the law. The AAA director had never read the law and did not know there was such an animal as the "grandfathering clause."
I am not the smartest person in the room but I have been studying what I am supposed to be doing so I know what the law says, and she had not, which means that I constitute an irritation to her because I can correct her when it comes down to what the law says.
From Edgar Puryear, George S. Brown, General, U.S. Air Force: Destined for Stars (1983) [combined snippets:
There were other occasions when he displayed sparks of temper. For example, he heard that an Air Force brigadier general was downgrading the A-10. Brown at that time was trying to work closely with the Army on the Air Force use of the A-10 for ground support. "He knew," said Dick Ellis, "that if the Army thought the Air Force was not serious about the coordination and usefulness of the A-10, he would be in trouble with Army Chief of Staff Creighton Abrams. I remember his telling me, 'Get that brigadier up here and I'll have a good piece of him.' But this brigadier, when he came in to see the Chief, was the smartest guy in the room. He had done his homework and knew the details of his position. After the explanation was given of what had actually happened, George cooled off. After he left, George said to me, 'That kid makes sense, doesn't he?'"
And from Crain's Illinois Business, volumes 2–3 (1983[?]):
Two of those inventions—the Telestrator electronic graphics system and the Discon teleconferencing system—already have established Interand as one of the most promising high-tech firms in the United States. The Telestrator is an ingenious briefcase-sized electronic extension of those wax slates children love to play with. "It's a way to reach into a television screen and draw on the picture," explains its 54-year-old inventor. He speaks with the self-deprecating simplicity of someone long accustomed to being the smartest person in the room. "The machine consists of a monitor covered by a screen, plus an electronic pen, which is connected to a computerized symbol-and-effects generator. A child can use it to animate an entire scene instantly."
These examples suggest that by 1983, the expression "smartest guy/person in the room" was fairly well established in political/business/bureaucratic settings. It did not become a trope in popular culture, however, until it appeared in Broadcast News:
Paul Moore: It must be nice to always believe you know better, to always think you're the smartest person in the room.
Jane Craig: No. It's awful.
And of course, it became even more familiar when it provided the central idiom of the book by Bethany McLean & Peter Elkind, The Smartest Guys in the Room: The Amazing Rise and Scandalous Fall of Enron (2003), and the subsequent film documentary, Enron: The Smartest Guys in the Room (2005).