The Grease star, 72, had become a television sensation through his role as Vinnie Barbarino in the hit sitcom Welcome Back, Kotter when filming began on Saturday Night Fever in Brooklyn in 1977.
The film's director John Badham has now revealed thousands of fans mobbed its locations while he was battling to make the classic movie.
"On our first day of shooting, (fans) almost completely shut us down. In fact, they did shut us down by lunchtime," Badham told The Hollywood Reporter's It Happened in Hollywood podcast.
Recalling the chaos of filming on the streets of Brooklyn, Badham explained the production struggled to capture even simple scenes because of the enormous crowds gathering around its lead actor.
"We shot in the morning, but we had 15,000 people on the streets of Brooklyn getting in every camera shot that we had, and all we're trying to do is get a shot of (Travolta) buying a shirt in a shirt store," he said.
Fans recognised the actor not as his Saturday Night Fever character Tony Manero, but as his television alter ego.
"We were underneath the 86th Street train in Brooklyn, and these little girls were going up the steps to the elevated train. And they start calling out to John, who's standing below where we're lining up a shot, and they say, 'Vinnie Barbarino!' And John is waving to them."
What began as a handful of fans rapidly escalated into a crowd of thousands.
"About eight or 10 people" initially appeared before "there were 15,000 people out there struggling to see them", he said.
To stay ahead of the crowds, the crew resorted to increasingly inventive tactics, including creating false production schedules.
"We're having to open up on the street at 6.30 in the morning to shoot before people are awake and know that we're out of there, so we can run and hide in one of our interior sets like the paint store."
Eventually, however, they realised the crew was circulating fake call sheets.
"They had figured out the fake call sheets, and someone said, 'I tell you what, instead of one car that we have a camera on, let's get this double. We have two identical Travolta cars, and we'll pile people in it and send them off like that's the real shooting car, and then we'll take the real one in a different direction'."
Reflecting on the performance that made Travolta an international icon, Badham said the actor understood Tony Manero better than anyone.
"He understood that character way better than me. John understood exactly what that character was about and where he was coming from and what he thought about himself."
Saturday Night Fever went on to become one of the defining cultural phenomena of the 1970s, transforming Travolta into a global superstar and cementing his place in cinema history, alongside later successes including Grease, Pulp Fiction, Face/Off and Hairspray.
The soundtrack by the Bee Gees included songs: Stayin' Alive, Night Fever, More Than a Woman and How Deep Is Your Love, which  became synonymous with the disco era and helped make Saturday Night Fever and its soundtrack a worldwide phenomenon.
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