“Searching” (initially titled “Search”) is one of those movies, like “Tarnation” or “The Blair Witch Project,” so unique in its approach that Sundance can only program something of its kind once before the gimmick gets old — in this case, a taut missing-persons investigation that unfolds entirely on computer screens. Good thing then that first-time feature director Aneesh Chaganty has concocted such an exceptionally clever riff on that idea. Cutting to the emotional core of what social media says about us, the result is as much a time capsule of our relationship to (and reliance upon) modern technology as it is a cutting-edge digital thriller.

No wonder then that Sony shelled out $5 million for worldwide rights to a film that others will surely be imitating for years to come. Not that Chaganty invented the big-screen-as-computer-screen conceit — that approach has already been quite lucrative for Timur Bekmambetov, who produced both “Searching” and the “Unfriended” franchise, with countless others trying similar stunts. Here, the genre element should get audiences in the door, rewarding them with twist after twist as “Searching” unfolds, though Chaganty’s smartest move was to ground the story in something so human as a strong father-daughter relationship.

Booting up with a 15-year tour through the Kim family’s recent past, “Searching” rivals the opening montage of Pixar’s “Up” for sheer heart-wrenching impact. This bravura sequence zooms into relevant portions of the screen as proud dad David (John Cho) pilots the cursor, sampling from YouTube videos, email messages, calendar appointments, etc., to recap the early milestones in the life of daughter Margot (played by Michelle La in the present). In a nice touch, a simple melody from her early piano recitals provides the music, which turns melancholy when mom Pamela (Sara Sohn) is diagnosed with a terminal case of lymphoma.

Feeling: teary-eyed emoji. Seldom can a film put audiences through such a roller-coaster of emotions within the span of its first few minutes, and yet Chaganty earns what might have felt like manipulation in someone else’s hands, using Pamela’s death to inform the concerned-parent dynamic David, now a widower, shows toward his daughter, now a 16-year-old high school student — old enough to keep secrets, but so responsible he rarely has to worry where she is. Until the night she disappears.

“Searching” parcels out clues so discretely we hardly recognize them as such (except for the in-joke “Home of the Catfish” sign glimpsed in the background of one family photo). Instead, familiar conversations about daily chores and establishing curfew assume added significance only after Margot fails to come home one night. Everything significant happens while the Apple “flurry” screensaver floats jellyfish-like across a dark computer screen: David misses two calls and an invitation to FaceTime from Margot, and when he wakes up the next morning, it takes him a while to realize something’s wrong.

And then he starts to freak out, blowing up her phone with nagging-dad texts (the same messages come flooding back, simultaneously amusing and ominous, when he logs on to her computer). David calls his pothead brother Peter (Joseph Lee) for advice, then begins to do precisely what no teenager wants: He reaches out to each of Margot’s “friends,” demanding to know when they last saw her. His daughter would be mortified — if she’s not already dead, that is.

Detective Rosemary Vick (Debra Messing) is optimistic. She specializes in missing-children cases, and volunteers to help, doubling as a sounding board for David’s many wild theories on what could have happened. At first, Chaganty finds it relatively easy to track the progress of David’s investigation on-screen. His actions are logical, as he moves from one social-media platform to another — Instagram, Facebook, a Twitch-like self-streaming site called YouCast — but he’s also a dad, and some of these sites are new to him. “What’s a tumbler?” he asks at one point.

Stumped by a password he can’t guess, David uses a sneaky trick to hack Margot’s account, only to discover that his daughter isn’t nearly as popular as he imagined. In fact, since her mom’s death, Margot may have been leading a double life, stealing money, and communicating with strangers online. Suddenly, the mystery is bigger than who may have kidnapped Margot. David desperately wants to know what’s been going on in her head. (Maybe he should check her Kindle to see if she’s been reading “Gone Girl.”)

Though he occasionally jumps from one computer to another, Chaganty sticks to the computer-screen rule, although a weird thing happens two-thirds of the way into the movie: At a certain point, it’s no longer clear who is actually operating the machine, as the suddenly-omniscient storyteller watches the juiciest part of the story unfold on YouTube and various local news sites: David drives out to Barbosa Lake (cue GPS app), confronts a disrespectful classmate (the skirmish is caught on a bystander’s iPhone and uploaded to YouTube), and installs hidden cameras in another suspect’s apartment (we see the feed he intends to use as evidence).

By this point in the case, it’s not clear why we’re still stuck in the computer. After all, “Searching” has talented actors in Cho and Messing. Wouldn’t a traditional format have been more effective than squinting at them in various low-res online clips? (As possible models, both “District 9” and “End of Watch” began as found-footage movies, but expanded to include non-diegetic cameras when their stories heated up.) One possible solution: By sticking to his conceit, Chaganty can comment on a fascinating new phenomenon — namely, how the “court of public opinion” swiftly leaps to its own conclusions.

On Twitter, the case goes viral, and the #FindMargot hashtag brings out the hypocrisy in everyone: Classmates who hardly knew her claim to have been Margot’s best friend, soaking up the sympathy intended for her, while haters rush to assume #DadDidIt. But until her body reappears, dead or alive, the mystery remains compellingly difficult to crack, with new revelations forcing both David and the audience to reconsider everything right up until the final reveal.

At Sundance, the film was projected for an audience, whose collective gasps and laughter fuel the experience, but of course, many will watch “Searching” alone on their personal computer screens — which Chaganty must have had in mind, when designing how to frame the action. He seldom shows David’s full screen, but instead, crops and cuts in such a way as to emphasize where we should be looking (which isn’t always where the cursor is pointing). Editors Will Merrick and Nick Johnson deserve special credit for assembling a complex 3D puzzle that seems to be happening in real time, creating both urgency and the illusion that we have an active role in solving it, while actually forcing us down a predetermined path, full of red herrings and uncanny suspense sequences.

None of this would matter if we didn’t care about the characters, and in “Searching,” Chaganty has found a new idiom for communicating not only the things we share, but also those we keep hidden from the ones we love.