Stewards said in a statement that the turn one incident, during the afternoon session at the Circuit de Catalunya, was considered "a significant breach of the safety requirements".
They also asked the governing FIA safety delegate to ensure the Toro Rosso cars complied with the technical regulations before Saturday's third session.
Red Bull's Daniel Ricciardo was handed a similar 10-place penalty at last month's Bahrain Grand Prix after his car shed a front left wheel following a pitstop during the March race in Malaysia.
Toro Rosso are seventh in the constructors' standings with eight points from five races.
Vergne, whose team mate is Russian rookie Daniil Kvyat, has scored half of the points.
($1 = 0.7269 Euros)
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