Ecclestone, ousted as the sport's commercial supremo in January after Liberty Media completed their takeover, was famously sceptical about social media and highly controlling of all video content from inside the paddock.
Much of Formula One's revenues comes from television deals with broadcasters.
Liberty Media have made fan engagement a priority, however, and are determined to build the brand through social media and digital channels.
Teams were told in guidelines issued ahead of the first day of pre-season testing in Barcelona on Monday that they could now "shoot short-form video for use on the team's own social channels" from inside the paddock.
It added that a formal set of content guidelines would be issued in due course.
The new policy was immediately evident, with Mercedes' triple world champion Lewis Hamilton - who has 3.8 million followers on Instagram and 4.16 million on Twitter - promising more access.
"Hey guys. I'm about to get on the car. Through practice I'll be live on Instagram from the cockpit to give you something new, so tune in," he said on Twitter.
The coverage consisted of footage shot in the pit lane and garage, but not on the track, and lasting no more than 10 seconds.
Red Bull team principal Christian Horner said fans would now have more behind-the-scenes access to drivers.
"You've probably noticed the teams are putting out a bit more content of what their drivers are up to in the paddock, not what's happening trackside because that belongs to the broadcasters," he told reporters.
"You could have had breakfast with (Australian) Daniel Ricciardo this morning, seen what he does prior to getting in the car and to the point where he's got in the car to drive out and do his first lap.
"Which is something that obviously hasn't been possible in previous years."
Horner said Ecclestone's departure from a hands-on role, with the Briton now handed the title of "Chairman Emeritus", would take some getting used to.
The new owners had a clear vision of what they wanted, however.
"They seem very keen to embrace ideas to make the sport more accessible and entertaining," said Horner. "In the immediacy, what we can see is that Formula One will perhaps be made slightly more accessible."