The Children of Men actor was at the international movie festival to promote his new HBO film Hemingway & Gellhorn, in which he stars opposite Nicole Kidman.
According to The Sun, Owen and his two young daughters Hannah and Eve were trapped in an elevator at the Hotel Majestic Barriere.
Two other people stuck with them fainted by the time they were freed.
Owen was forced to cancel numerous press interviews in order to calm his daughters, who were visibly shaken.
When told of the incident by reporters, co-star Kidman said: "I couldn't have been in there with my kids, how awful."
Owen will star alongside Mila Kunis and Marion Cotillard in the upcoming thriller Blood Ties.
Watch a clip from Hemingway & Gellhorn below: