Leclerc passed Vettel with the benefit of fresh tyres shortly after the safety-car period necessitated by Valtteri Bottas' retirement, but the German came back immediately with the aid of DRS and repassed Leclerc on the back straight.
However, Vettel clipped wheels with Leclerc as he made his way through, causing a dramatic failure on Leclerc's right-front suspension, and picking up a puncture of his own.
Both immediately had to retire, making the Interlagos race Ferrari's first scoreless outing since the 2017 Singapore GP.
Elkann suggested that teamwork had gone out the window, with Vettel and Leclerc given freedom to fight on-track by team principal Mattia Binotto, who also earned the backing of Elkann over the incident.
"What happened on Sunday not only made me angry, what happened in Brazil made us realize how important Ferrari is," Elkann told an Exor investors' call.
"It must not happen that the drivers, no matter how good they are, forget that they are always Ferrari drivers.
"What matters is the teamwork and that Ferrari wins, on this aspect Binotto has always been very clear.
"It is a positive season for Ferrari if you look at the number of poles, which was not so high since 1970, unfortunately we were not able to convert pole positions into victories."
Binotto summoned Vettel and Leclerc to Maranello to discuss the incident, having barred the pair from post-race debriefs.
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