When it comes to crazy plans, there’s a fine line between “just crazy enough to work” and “so crazy it won’t work.” That line is explored to great effect in “Argo,” a supremely entertaining and stimulating film, directed by Ben Affleck, about the time the CIA used a fake movie production to try to sneak Americans out of Iran during the 1979-81 hostage crisis. Objectively speaking, it was a bad idea. But as one of the characters in the movie puts it, “This is the best bad idea we have.”

The mission took place in January 1980 and was big news at the time, but the amazing account of what really happened wasn’t declassified until 1997. By then, the episode was no longer of general interest, and it had been overshadowed in history by the release of the 52 hostages in January 1981. The semi-obscurity works to the movie’s advantage, as the average viewer under the age of about 40 probably hasn’t heard the story. The terrific suspense that Affleck achieves in the third act is all the more potent when you don’t know how it’s going to end.

In November 1979, when the American embassy in Tehran is seized by Iranian militants and more than 50 people are taken hostage, six U.S. diplomats manage to escape to the home of the Canadian ambassador (played here by Victor Garber). Over the next several weeks, as the world watches the crisis unfold and the U.S. government negotiates for the hostages’ release, the State Department secretly looks for ways to get these other six out of the country. They’re safe for now — but only because the terrorists don’t know they exist. Once the militants realize they’re missing a half-dozen Americans, they’ll tear Tehran apart looking for them.

Time is of the essence, therefore, when the CIA calls in “exfiltration” specialist Tony Mendez (Affleck). Sneaking across the border is out of the question; the only realistic way to get the fugitives out of Iran safely is to give them fake identities and cover stories and let them take a commercial flight like anybody else. After pointing out the reasons why all of the higher-ups’ ideas won’t pass scrutiny with Iranian officials (there are very few plausible reasons for Westerners to be in Tehran at this point), Tony comes up with the aforementioned crazy plan: the six will pass themselves off as Hollywood-backed Canadian filmmakers who are in Iran scouting locations for a movie.

Here we shift gears smoothly from a serious-minded (but not humorless) political docudrama to something like a Hollywood satire. To make the fake movie production seem authentic, Tony enlists jovial Hollywood makeup artist John Chambers (John Goodman) and past-his-prime producer Lester Siegel (Alan Arkin) to help him fill in the details. It has to look right, not just for the sake of the six diplomats’ cover story but for the sake of Siegel’s reputation: “If I’m doing a fake movie, it’s gonna be a fake hit!” They find a screenplay called “Argo” that would require desert settings (it’s a “Star Wars” rip-off), set up a production company, and commission a poster. Having once wandered through the film market at Cannes, I can tell you that a title and a poster are pretty much all you need to convince people you’re making a movie.

These Hollywood scenes are very funny without devolving into silly parody. Chris Terrio’s screenplay wisely avoids making wink-wink inside jokes, focusing instead on industry-related humor that outsiders can appreciate. I kept bracing myself for wacky cameos; fortunately, none came. If there’s maybe one too many gags along the lines of “You’re worried about the Ayatollah? Try the WGA!,” it’s easily forgivable in light of Arkin and Goodman’s cheerful performances as seasoned, lightly cynical showbiz veterans. They’re scene-stealers, in the best possible way.

Yet even here, in this comical middle section, Affleck deftly keeps the film thematically cohesive, and also avoids undercutting the life-or-death seriousness of the situation. To give the fake “Argo” credibility, Siegel stages an event that will get it written up in the trade papers; as the old saying goes, if you want to sell a lie, get the press to sell it for you. Meanwhile, in Tehran, the revolutionaries put on their own kind of show, giving impassioned speeches in front of news cameras so that their cause will remain in the public consciousness. While there are varying degrees of sincerity involved, both sides are using the media to get their message across.

With the plan now in place, Affleck shifts gears again, and “Argo” becomes a tight, breathless suspense film. Tony must handle the nuts and bolts of giving the six fugitives their cover stories, schooling them in how to pretend to be filmmakers in a way that will convince the authorities, and actually getting them on a plane. The whole charade could collapse at any moment, especially given how nervous the six are about its audacity.

Drawing from a Wired magazine article by Joshuah Bearman and a book by the real Tony Mendez, “Argo’s” screenplay is a marvel of efficiency. (One downside: Canada’s role in the affair is drastically understated.) The film is two hours long and extremely plot-heavy, yet it’s densely packed with just enough information about the many, many characters to keep us interested in them. Tony has a son he adores and a wife from whom he is separated, but that angle isn’t overplayed. He has a brief but memorable conversation with Lester Siegel about their respective families that brings both characters to life in only a few lines, humanizing them without being schmaltzy. Reliable actors like Bryan Cranston, Kyle Chandler, Bob Gunton, and Philip Baker Hall, playing various CIA and State Department officials, bark their caustic, profanity-laced dialogue with gusto, like a less satiric “In the Loop.” As the fugitives, Clea DuVall, Christopher Denham, Tate Donovan, Rory Cochrane, Kerry Bishé, and Scoot McNairy each get at least a couple moments to establish themselves.

If Affleck’s bonafides as a director were still in question after “Gone Baby Gone” and “The Town,” “Argo” should be the end of that. With an impressive eye for period and location details (and he’s not on his home turf anymore, either), Affleck guides this fraught, multi-faceted story with the confident and un-showy skill of a man who has directed thirteen movies, not three. This is mature, invigorating stuff, alternately tense, funny, stirring, and ultimately very satisfying.

A- (2 hrs.; R, a lot of harsh profanity, some violent images.)

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