East, 35, ended up triumphing and went through to next week while Bye was handed his marching orders in the results show, which was pre-recorded.
The Sun reported that East was meant to pull Coppola along in a prop chair and that the legs slipped and she fell.
She was then allowed to re-start.
The BBC said the restart had been permitted because the pair had “not begun to dance” before the incident occurred.
In a statement shared with the PA news agency, a BBC spokesperson said the show had been stopped “as a matter of urgency”.
“After an incident with a prop at the beginning of Fleur and Vito’s dance-off performance, the decision was made to halt proceedings to check they were not injured, as a matter of urgency,” the spokesperson said.
“As they had not begun to dance before the incident occurred, it was decided they could start the performance again, once it was confirmed they were fit to do so.”
Mabuse said there were “different levels of dancing on the dancefloor”.
And Revel Horwood said it was his “responsibility” as a judge to “save the better dancer”.
In the end all four chose to save East and Coppola.
Strictly returns on Saturday at 7pm, with the results show on Sunday at 7:15pm on BBC One.
Additional reporting from PA.
Watch: Fleur East 'redoes' dance performance on Strictly after falling on partner Vito Coppola